How challenging did you find Real Analysis before the penny dropped?












2














I've been following 'Understanding Analysis by Stephen Abbot' and have been struggling with exercise questions even after an in depth read of the chapters. I have read numerous statements by authors that 'if you can't do the exercises, then you haven't grasped the material properly', but I feel like I have grasped the concepts. I've read many 'how to' books to finally understand higher level math and persevered with a 'never giving up' attitude, but most exercises really throw me off. Even those in the first chapter which are meant to be on basic preliminary material.



Is this just the nature of studying Real Analysis? I'll be starting Abstract Algebra soon and have been advised to purchase 'J.B Fraleighs introduction'. Do other higher maths subjects cause the same anxieties as Real Analysis?










share|cite|improve this question






















  • Could you give some specific examples of problems that you find difficult?
    – Sambo
    5 hours ago










  • It seems what you’re really looking for is commiseration in your time of trouble, which really isn’t a mathematical question. But I’m sure you’d find commiseration in the chatrooms...
    – rschwieb
    5 hours ago






  • 3




    You need to read Rudin. It's the only respectable analysis text, in my honest view. Only by reading baby Rudin (the bible of undergraduate mathematics) will you truly begin to learn analysis.
    – MathematicsStudent1122
    5 hours ago








  • 1




    I've not long started either, but I'd say there's often a difference between grasping concepts and doing exercises—namely that an exercise can often require some useful little trick which isn't mentioned in the material: some algebraic manoeuvre, say. (And the author isn't writing about algebra, so leaves that part to you.)
    – timtfj
    5 hours ago






  • 3




    MathematicsStudent is right, Baby Rudin is a really good place to jump in. Later, you meet Papa Rudin and Grandpa Rudin as well! ;)
    – Robert Lewis
    5 hours ago
















2














I've been following 'Understanding Analysis by Stephen Abbot' and have been struggling with exercise questions even after an in depth read of the chapters. I have read numerous statements by authors that 'if you can't do the exercises, then you haven't grasped the material properly', but I feel like I have grasped the concepts. I've read many 'how to' books to finally understand higher level math and persevered with a 'never giving up' attitude, but most exercises really throw me off. Even those in the first chapter which are meant to be on basic preliminary material.



Is this just the nature of studying Real Analysis? I'll be starting Abstract Algebra soon and have been advised to purchase 'J.B Fraleighs introduction'. Do other higher maths subjects cause the same anxieties as Real Analysis?










share|cite|improve this question






















  • Could you give some specific examples of problems that you find difficult?
    – Sambo
    5 hours ago










  • It seems what you’re really looking for is commiseration in your time of trouble, which really isn’t a mathematical question. But I’m sure you’d find commiseration in the chatrooms...
    – rschwieb
    5 hours ago






  • 3




    You need to read Rudin. It's the only respectable analysis text, in my honest view. Only by reading baby Rudin (the bible of undergraduate mathematics) will you truly begin to learn analysis.
    – MathematicsStudent1122
    5 hours ago








  • 1




    I've not long started either, but I'd say there's often a difference between grasping concepts and doing exercises—namely that an exercise can often require some useful little trick which isn't mentioned in the material: some algebraic manoeuvre, say. (And the author isn't writing about algebra, so leaves that part to you.)
    – timtfj
    5 hours ago






  • 3




    MathematicsStudent is right, Baby Rudin is a really good place to jump in. Later, you meet Papa Rudin and Grandpa Rudin as well! ;)
    – Robert Lewis
    5 hours ago














2












2








2


2





I've been following 'Understanding Analysis by Stephen Abbot' and have been struggling with exercise questions even after an in depth read of the chapters. I have read numerous statements by authors that 'if you can't do the exercises, then you haven't grasped the material properly', but I feel like I have grasped the concepts. I've read many 'how to' books to finally understand higher level math and persevered with a 'never giving up' attitude, but most exercises really throw me off. Even those in the first chapter which are meant to be on basic preliminary material.



Is this just the nature of studying Real Analysis? I'll be starting Abstract Algebra soon and have been advised to purchase 'J.B Fraleighs introduction'. Do other higher maths subjects cause the same anxieties as Real Analysis?










share|cite|improve this question













I've been following 'Understanding Analysis by Stephen Abbot' and have been struggling with exercise questions even after an in depth read of the chapters. I have read numerous statements by authors that 'if you can't do the exercises, then you haven't grasped the material properly', but I feel like I have grasped the concepts. I've read many 'how to' books to finally understand higher level math and persevered with a 'never giving up' attitude, but most exercises really throw me off. Even those in the first chapter which are meant to be on basic preliminary material.



Is this just the nature of studying Real Analysis? I'll be starting Abstract Algebra soon and have been advised to purchase 'J.B Fraleighs introduction'. Do other higher maths subjects cause the same anxieties as Real Analysis?







soft-question book-recommendation advice






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked 5 hours ago









user503154

823




823












  • Could you give some specific examples of problems that you find difficult?
    – Sambo
    5 hours ago










  • It seems what you’re really looking for is commiseration in your time of trouble, which really isn’t a mathematical question. But I’m sure you’d find commiseration in the chatrooms...
    – rschwieb
    5 hours ago






  • 3




    You need to read Rudin. It's the only respectable analysis text, in my honest view. Only by reading baby Rudin (the bible of undergraduate mathematics) will you truly begin to learn analysis.
    – MathematicsStudent1122
    5 hours ago








  • 1




    I've not long started either, but I'd say there's often a difference between grasping concepts and doing exercises—namely that an exercise can often require some useful little trick which isn't mentioned in the material: some algebraic manoeuvre, say. (And the author isn't writing about algebra, so leaves that part to you.)
    – timtfj
    5 hours ago






  • 3




    MathematicsStudent is right, Baby Rudin is a really good place to jump in. Later, you meet Papa Rudin and Grandpa Rudin as well! ;)
    – Robert Lewis
    5 hours ago


















  • Could you give some specific examples of problems that you find difficult?
    – Sambo
    5 hours ago










  • It seems what you’re really looking for is commiseration in your time of trouble, which really isn’t a mathematical question. But I’m sure you’d find commiseration in the chatrooms...
    – rschwieb
    5 hours ago






  • 3




    You need to read Rudin. It's the only respectable analysis text, in my honest view. Only by reading baby Rudin (the bible of undergraduate mathematics) will you truly begin to learn analysis.
    – MathematicsStudent1122
    5 hours ago








  • 1




    I've not long started either, but I'd say there's often a difference between grasping concepts and doing exercises—namely that an exercise can often require some useful little trick which isn't mentioned in the material: some algebraic manoeuvre, say. (And the author isn't writing about algebra, so leaves that part to you.)
    – timtfj
    5 hours ago






  • 3




    MathematicsStudent is right, Baby Rudin is a really good place to jump in. Later, you meet Papa Rudin and Grandpa Rudin as well! ;)
    – Robert Lewis
    5 hours ago
















Could you give some specific examples of problems that you find difficult?
– Sambo
5 hours ago




Could you give some specific examples of problems that you find difficult?
– Sambo
5 hours ago












It seems what you’re really looking for is commiseration in your time of trouble, which really isn’t a mathematical question. But I’m sure you’d find commiseration in the chatrooms...
– rschwieb
5 hours ago




It seems what you’re really looking for is commiseration in your time of trouble, which really isn’t a mathematical question. But I’m sure you’d find commiseration in the chatrooms...
– rschwieb
5 hours ago




3




3




You need to read Rudin. It's the only respectable analysis text, in my honest view. Only by reading baby Rudin (the bible of undergraduate mathematics) will you truly begin to learn analysis.
– MathematicsStudent1122
5 hours ago






You need to read Rudin. It's the only respectable analysis text, in my honest view. Only by reading baby Rudin (the bible of undergraduate mathematics) will you truly begin to learn analysis.
– MathematicsStudent1122
5 hours ago






1




1




I've not long started either, but I'd say there's often a difference between grasping concepts and doing exercises—namely that an exercise can often require some useful little trick which isn't mentioned in the material: some algebraic manoeuvre, say. (And the author isn't writing about algebra, so leaves that part to you.)
– timtfj
5 hours ago




I've not long started either, but I'd say there's often a difference between grasping concepts and doing exercises—namely that an exercise can often require some useful little trick which isn't mentioned in the material: some algebraic manoeuvre, say. (And the author isn't writing about algebra, so leaves that part to you.)
– timtfj
5 hours ago




3




3




MathematicsStudent is right, Baby Rudin is a really good place to jump in. Later, you meet Papa Rudin and Grandpa Rudin as well! ;)
– Robert Lewis
5 hours ago




MathematicsStudent is right, Baby Rudin is a really good place to jump in. Later, you meet Papa Rudin and Grandpa Rudin as well! ;)
– Robert Lewis
5 hours ago










3 Answers
3






active

oldest

votes


















8














I have two insights I have gleaned from my personal struggle with similar difficulties:



First, if you are having trouble with exercises, find easier exercises on which to cut your teeth. There are always easier exercises. The point is to jump in at a place where you feel comfortable working so you can grow your skills naturally. You may have to switch books to do it, but it can be done. Skill grows inevitably with practice, but I think you have to practice with what you can do. Sometimes baby steps are the wisest way to go. Just my thoughts . . .



Second, in my experience as a student, teacher, and researcher, some people are just naturally better at some subjects than they are at others. I have seen people who could breeze through graduate algebra courses struggle to barely pass analysis or differential equations, and vice versa; and I think putting time in on what you have a natural knack for may contribute to your success in subjects which you find intrinsically more difficult.



I hope these words are helpful. You are not alone.






share|cite|improve this answer

















  • 1




    Just to add to your comment that "people are just naturally better at some subjects than they are at other" there is also the possibility that you might 'naturally' struggle with real analysis, but find topology 'naturally' accessible, which will then make real analysis intuitive. Sometimes pushing through with a bare understanding of the main results and then going back works.
    – Bernard W
    4 hours ago










  • @BernardW: yes; I couldn't really grasp Lebesgue integration very well until I learned the essentials of functional analysis. Go figure!
    – Robert Lewis
    4 hours ago



















5














I would say that the ability to do the exercises is an indication that you have (or are currently) mastering the concepts, but that simply grasping them is a much lower bar.



My personal advice is the following: find some classmates and take turns teaching each other the material. Literally. Go find a study room or somewhere you can stand in front of a board and teach each other what you learned in class. Reprove theorems. Re-examine examples. Re-state definitions (from memory if you can). There is no better way to learn than to teach.



If that doesn't help, then my next bit of advice is to practice 'till you puke. Another answer stated something about easier exercises to cut your teeth; this is an amazing bit of advice. But don't feel compelled to get back up to the hard problems fast! Keep practicing everything, because you never know when it will all click, and what will be the cause of said click.






share|cite|improve this answer








New contributor




DeficientMathDude is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.


























    0














    A few bits of advice...



    First, draw pictures. Specifically, draw graphs of real-valued functions, and see what various theorems are telling you about these pictures. Draw pictures to help you solve problems, too. Pictures don’t constitute a solution or a proof, but they can often guide you towards the right line of reasoning.



    Second, play with lots of examples. Suppose you’re studying a theorem that has several hypotheses. Think of examples where the hypotheses are satisfied and where they are not. If one of the hypotheses is not satisfied, maybe the theorem is no longer true. Think of examples that illustrate this.



    Third, don’t expect things to be easy. Mathematics texts (and many teachers) have a nasty habit of showing you only the pretty final results, without showing you all the false starts and roadblocks that preceded them. You won’t be able to write textbook mathematics on your first attempt. The first few attempts will be a mess. Once you have a solution that works, then you can tidy it up and make it look as pretty as the stuff in textbooks.






    share|cite|improve this answer





















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      3 Answers
      3






      active

      oldest

      votes








      3 Answers
      3






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      8














      I have two insights I have gleaned from my personal struggle with similar difficulties:



      First, if you are having trouble with exercises, find easier exercises on which to cut your teeth. There are always easier exercises. The point is to jump in at a place where you feel comfortable working so you can grow your skills naturally. You may have to switch books to do it, but it can be done. Skill grows inevitably with practice, but I think you have to practice with what you can do. Sometimes baby steps are the wisest way to go. Just my thoughts . . .



      Second, in my experience as a student, teacher, and researcher, some people are just naturally better at some subjects than they are at others. I have seen people who could breeze through graduate algebra courses struggle to barely pass analysis or differential equations, and vice versa; and I think putting time in on what you have a natural knack for may contribute to your success in subjects which you find intrinsically more difficult.



      I hope these words are helpful. You are not alone.






      share|cite|improve this answer

















      • 1




        Just to add to your comment that "people are just naturally better at some subjects than they are at other" there is also the possibility that you might 'naturally' struggle with real analysis, but find topology 'naturally' accessible, which will then make real analysis intuitive. Sometimes pushing through with a bare understanding of the main results and then going back works.
        – Bernard W
        4 hours ago










      • @BernardW: yes; I couldn't really grasp Lebesgue integration very well until I learned the essentials of functional analysis. Go figure!
        – Robert Lewis
        4 hours ago
















      8














      I have two insights I have gleaned from my personal struggle with similar difficulties:



      First, if you are having trouble with exercises, find easier exercises on which to cut your teeth. There are always easier exercises. The point is to jump in at a place where you feel comfortable working so you can grow your skills naturally. You may have to switch books to do it, but it can be done. Skill grows inevitably with practice, but I think you have to practice with what you can do. Sometimes baby steps are the wisest way to go. Just my thoughts . . .



      Second, in my experience as a student, teacher, and researcher, some people are just naturally better at some subjects than they are at others. I have seen people who could breeze through graduate algebra courses struggle to barely pass analysis or differential equations, and vice versa; and I think putting time in on what you have a natural knack for may contribute to your success in subjects which you find intrinsically more difficult.



      I hope these words are helpful. You are not alone.






      share|cite|improve this answer

















      • 1




        Just to add to your comment that "people are just naturally better at some subjects than they are at other" there is also the possibility that you might 'naturally' struggle with real analysis, but find topology 'naturally' accessible, which will then make real analysis intuitive. Sometimes pushing through with a bare understanding of the main results and then going back works.
        – Bernard W
        4 hours ago










      • @BernardW: yes; I couldn't really grasp Lebesgue integration very well until I learned the essentials of functional analysis. Go figure!
        – Robert Lewis
        4 hours ago














      8












      8








      8






      I have two insights I have gleaned from my personal struggle with similar difficulties:



      First, if you are having trouble with exercises, find easier exercises on which to cut your teeth. There are always easier exercises. The point is to jump in at a place where you feel comfortable working so you can grow your skills naturally. You may have to switch books to do it, but it can be done. Skill grows inevitably with practice, but I think you have to practice with what you can do. Sometimes baby steps are the wisest way to go. Just my thoughts . . .



      Second, in my experience as a student, teacher, and researcher, some people are just naturally better at some subjects than they are at others. I have seen people who could breeze through graduate algebra courses struggle to barely pass analysis or differential equations, and vice versa; and I think putting time in on what you have a natural knack for may contribute to your success in subjects which you find intrinsically more difficult.



      I hope these words are helpful. You are not alone.






      share|cite|improve this answer












      I have two insights I have gleaned from my personal struggle with similar difficulties:



      First, if you are having trouble with exercises, find easier exercises on which to cut your teeth. There are always easier exercises. The point is to jump in at a place where you feel comfortable working so you can grow your skills naturally. You may have to switch books to do it, but it can be done. Skill grows inevitably with practice, but I think you have to practice with what you can do. Sometimes baby steps are the wisest way to go. Just my thoughts . . .



      Second, in my experience as a student, teacher, and researcher, some people are just naturally better at some subjects than they are at others. I have seen people who could breeze through graduate algebra courses struggle to barely pass analysis or differential equations, and vice versa; and I think putting time in on what you have a natural knack for may contribute to your success in subjects which you find intrinsically more difficult.



      I hope these words are helpful. You are not alone.







      share|cite|improve this answer












      share|cite|improve this answer



      share|cite|improve this answer










      answered 5 hours ago









      Robert Lewis

      43.2k22863




      43.2k22863








      • 1




        Just to add to your comment that "people are just naturally better at some subjects than they are at other" there is also the possibility that you might 'naturally' struggle with real analysis, but find topology 'naturally' accessible, which will then make real analysis intuitive. Sometimes pushing through with a bare understanding of the main results and then going back works.
        – Bernard W
        4 hours ago










      • @BernardW: yes; I couldn't really grasp Lebesgue integration very well until I learned the essentials of functional analysis. Go figure!
        – Robert Lewis
        4 hours ago














      • 1




        Just to add to your comment that "people are just naturally better at some subjects than they are at other" there is also the possibility that you might 'naturally' struggle with real analysis, but find topology 'naturally' accessible, which will then make real analysis intuitive. Sometimes pushing through with a bare understanding of the main results and then going back works.
        – Bernard W
        4 hours ago










      • @BernardW: yes; I couldn't really grasp Lebesgue integration very well until I learned the essentials of functional analysis. Go figure!
        – Robert Lewis
        4 hours ago








      1




      1




      Just to add to your comment that "people are just naturally better at some subjects than they are at other" there is also the possibility that you might 'naturally' struggle with real analysis, but find topology 'naturally' accessible, which will then make real analysis intuitive. Sometimes pushing through with a bare understanding of the main results and then going back works.
      – Bernard W
      4 hours ago




      Just to add to your comment that "people are just naturally better at some subjects than they are at other" there is also the possibility that you might 'naturally' struggle with real analysis, but find topology 'naturally' accessible, which will then make real analysis intuitive. Sometimes pushing through with a bare understanding of the main results and then going back works.
      – Bernard W
      4 hours ago












      @BernardW: yes; I couldn't really grasp Lebesgue integration very well until I learned the essentials of functional analysis. Go figure!
      – Robert Lewis
      4 hours ago




      @BernardW: yes; I couldn't really grasp Lebesgue integration very well until I learned the essentials of functional analysis. Go figure!
      – Robert Lewis
      4 hours ago











      5














      I would say that the ability to do the exercises is an indication that you have (or are currently) mastering the concepts, but that simply grasping them is a much lower bar.



      My personal advice is the following: find some classmates and take turns teaching each other the material. Literally. Go find a study room or somewhere you can stand in front of a board and teach each other what you learned in class. Reprove theorems. Re-examine examples. Re-state definitions (from memory if you can). There is no better way to learn than to teach.



      If that doesn't help, then my next bit of advice is to practice 'till you puke. Another answer stated something about easier exercises to cut your teeth; this is an amazing bit of advice. But don't feel compelled to get back up to the hard problems fast! Keep practicing everything, because you never know when it will all click, and what will be the cause of said click.






      share|cite|improve this answer








      New contributor




      DeficientMathDude is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.























        5














        I would say that the ability to do the exercises is an indication that you have (or are currently) mastering the concepts, but that simply grasping them is a much lower bar.



        My personal advice is the following: find some classmates and take turns teaching each other the material. Literally. Go find a study room or somewhere you can stand in front of a board and teach each other what you learned in class. Reprove theorems. Re-examine examples. Re-state definitions (from memory if you can). There is no better way to learn than to teach.



        If that doesn't help, then my next bit of advice is to practice 'till you puke. Another answer stated something about easier exercises to cut your teeth; this is an amazing bit of advice. But don't feel compelled to get back up to the hard problems fast! Keep practicing everything, because you never know when it will all click, and what will be the cause of said click.






        share|cite|improve this answer








        New contributor




        DeficientMathDude is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
        Check out our Code of Conduct.





















          5












          5








          5






          I would say that the ability to do the exercises is an indication that you have (or are currently) mastering the concepts, but that simply grasping them is a much lower bar.



          My personal advice is the following: find some classmates and take turns teaching each other the material. Literally. Go find a study room or somewhere you can stand in front of a board and teach each other what you learned in class. Reprove theorems. Re-examine examples. Re-state definitions (from memory if you can). There is no better way to learn than to teach.



          If that doesn't help, then my next bit of advice is to practice 'till you puke. Another answer stated something about easier exercises to cut your teeth; this is an amazing bit of advice. But don't feel compelled to get back up to the hard problems fast! Keep practicing everything, because you never know when it will all click, and what will be the cause of said click.






          share|cite|improve this answer








          New contributor




          DeficientMathDude is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
          Check out our Code of Conduct.









          I would say that the ability to do the exercises is an indication that you have (or are currently) mastering the concepts, but that simply grasping them is a much lower bar.



          My personal advice is the following: find some classmates and take turns teaching each other the material. Literally. Go find a study room or somewhere you can stand in front of a board and teach each other what you learned in class. Reprove theorems. Re-examine examples. Re-state definitions (from memory if you can). There is no better way to learn than to teach.



          If that doesn't help, then my next bit of advice is to practice 'till you puke. Another answer stated something about easier exercises to cut your teeth; this is an amazing bit of advice. But don't feel compelled to get back up to the hard problems fast! Keep practicing everything, because you never know when it will all click, and what will be the cause of said click.







          share|cite|improve this answer








          New contributor




          DeficientMathDude is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
          Check out our Code of Conduct.









          share|cite|improve this answer



          share|cite|improve this answer






          New contributor




          DeficientMathDude is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
          Check out our Code of Conduct.









          answered 5 hours ago









          DeficientMathDude

          1012




          1012




          New contributor




          DeficientMathDude is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
          Check out our Code of Conduct.





          New contributor





          DeficientMathDude is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
          Check out our Code of Conduct.






          DeficientMathDude is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
          Check out our Code of Conduct.























              0














              A few bits of advice...



              First, draw pictures. Specifically, draw graphs of real-valued functions, and see what various theorems are telling you about these pictures. Draw pictures to help you solve problems, too. Pictures don’t constitute a solution or a proof, but they can often guide you towards the right line of reasoning.



              Second, play with lots of examples. Suppose you’re studying a theorem that has several hypotheses. Think of examples where the hypotheses are satisfied and where they are not. If one of the hypotheses is not satisfied, maybe the theorem is no longer true. Think of examples that illustrate this.



              Third, don’t expect things to be easy. Mathematics texts (and many teachers) have a nasty habit of showing you only the pretty final results, without showing you all the false starts and roadblocks that preceded them. You won’t be able to write textbook mathematics on your first attempt. The first few attempts will be a mess. Once you have a solution that works, then you can tidy it up and make it look as pretty as the stuff in textbooks.






              share|cite|improve this answer


























                0














                A few bits of advice...



                First, draw pictures. Specifically, draw graphs of real-valued functions, and see what various theorems are telling you about these pictures. Draw pictures to help you solve problems, too. Pictures don’t constitute a solution or a proof, but they can often guide you towards the right line of reasoning.



                Second, play with lots of examples. Suppose you’re studying a theorem that has several hypotheses. Think of examples where the hypotheses are satisfied and where they are not. If one of the hypotheses is not satisfied, maybe the theorem is no longer true. Think of examples that illustrate this.



                Third, don’t expect things to be easy. Mathematics texts (and many teachers) have a nasty habit of showing you only the pretty final results, without showing you all the false starts and roadblocks that preceded them. You won’t be able to write textbook mathematics on your first attempt. The first few attempts will be a mess. Once you have a solution that works, then you can tidy it up and make it look as pretty as the stuff in textbooks.






                share|cite|improve this answer
























                  0












                  0








                  0






                  A few bits of advice...



                  First, draw pictures. Specifically, draw graphs of real-valued functions, and see what various theorems are telling you about these pictures. Draw pictures to help you solve problems, too. Pictures don’t constitute a solution or a proof, but they can often guide you towards the right line of reasoning.



                  Second, play with lots of examples. Suppose you’re studying a theorem that has several hypotheses. Think of examples where the hypotheses are satisfied and where they are not. If one of the hypotheses is not satisfied, maybe the theorem is no longer true. Think of examples that illustrate this.



                  Third, don’t expect things to be easy. Mathematics texts (and many teachers) have a nasty habit of showing you only the pretty final results, without showing you all the false starts and roadblocks that preceded them. You won’t be able to write textbook mathematics on your first attempt. The first few attempts will be a mess. Once you have a solution that works, then you can tidy it up and make it look as pretty as the stuff in textbooks.






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                  A few bits of advice...



                  First, draw pictures. Specifically, draw graphs of real-valued functions, and see what various theorems are telling you about these pictures. Draw pictures to help you solve problems, too. Pictures don’t constitute a solution or a proof, but they can often guide you towards the right line of reasoning.



                  Second, play with lots of examples. Suppose you’re studying a theorem that has several hypotheses. Think of examples where the hypotheses are satisfied and where they are not. If one of the hypotheses is not satisfied, maybe the theorem is no longer true. Think of examples that illustrate this.



                  Third, don’t expect things to be easy. Mathematics texts (and many teachers) have a nasty habit of showing you only the pretty final results, without showing you all the false starts and roadblocks that preceded them. You won’t be able to write textbook mathematics on your first attempt. The first few attempts will be a mess. Once you have a solution that works, then you can tidy it up and make it look as pretty as the stuff in textbooks.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 31 mins ago









                  bubba

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                  29.9k32986






























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