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I'm currently taking a course in Real Analysis that uses Principles of Mathematical Analysis by Rudin, and having a somewhat difficult time on tests.

I always read that notions like compactness, connectedness, and other things covered in Chapter 2 are super important, but I just don't recognize how they come into play in later chapters like the one on Integration or Differentiation. I think this is causing me to do poorly on tests, since I fail to see connections where I should.

Does anyone have tips for doing proofs in analysis, or just in general studying it?

gabe
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  • Do you understand some basic proofs involving compactness and connectedness? E.g. that a continuous function on a compact set has a maximum and minimum, and that a continuous function on a connected set satisfies the intermediate value property? – Noah Schweber Mar 17 '16 at 02:08
  • I am familiar with the results, but don't necessarily recall the proofs. When we were on that chapter, I don't recall having difficulty on those proofs. – gabe Mar 17 '16 at 02:13
  • Related: http://math.stackexchange.com/questions/614365/how-do-you-remember-theorems –  Mar 17 '16 at 02:16
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    I think then that's my initial suggestion - re-read those proofs, and come up with counterexamples if the hypotheses are dropped. They are the kind of result I think you should be able to reconstruct the proof of, if you really understand what compactness and connectedness are doing; so it's probably a good place to start. – Noah Schweber Mar 17 '16 at 02:19
  • I think Munkres treats these topics in hisTopology text, in a very step by step, readable way. – Matematleta Mar 17 '16 at 02:20
  • The is not whether you have difficulties with the proofs, instead the issue is whether you know how to express the proofs, very quickly and concisely, in the language of compacntess and connectedness. – Lee Mosher Mar 17 '16 at 02:34
  • Thank you all, I will review these proofs and try to pick out exactly where compactness and connectedness come into play. I am beginning to think I simply do not study math effectively. – gabe Mar 17 '16 at 02:36
  • Compactness is a generalisation of closed-and-boundedness, and connectedness generalises intervals. – Henricus V. Mar 17 '16 at 02:38

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