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This is my first time posting, so I apologize in advance if my question is inappropriate here. I wanted to know if it would be beneficial for me to review certain calculus topics before I take my first real analysis course. I have noticed that some calculus topics are involved in real analysis courses (e.g. sequences, series, definition of limit). If this is true, then what calculus topics would be helpful to review? If this is not the case, what self-study methods would most likely benefit me before I journey into proof-based math?

EDIT: I am completely new to proof-based math, so I am looking for a couple of self-study suggestions that would likely prepare me well for the rigor of real analysis. By self-study suggestions, I mean reviewing certain calculus topics (if it would be helpful), and/or certain titles of books, etc.

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welcome to MSE! –  DanZimm May 15 '13 at 6:44
    
is it safe to assume you mean non-proof-based self study suggestions? Otherwise Mark's answer below seems to address the question –  DanZimm May 16 '13 at 3:48
    
Not exactly. I am looking for both kinds. I am particularly looking for introductory proof-based self study suggestions. –  Gina32 May 16 '13 at 3:53
    
aight - Mark's answer seems pretty sufficient for that, are you simply looking for other opinions? –  DanZimm May 16 '13 at 4:01
    
yes, I would like some suggestions about books that give an introduction to proof-based math and real analysis. –  Gina32 May 16 '13 at 4:11
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Certainly, a solid understanding of limits would help.

By this, I don't just mean that you can look at $\lim_{x\rightarrow 1}(x^2-1)/(x-1)$ and be able to compute the answer. I mean that you have a picture in mind, understand how the $\varepsilon\mbox{-}\delta$ language relates to that picture and (ideally) are able to write down the $\varepsilon\mbox{-}\delta$ proof.

Beyond that, maybe the way you think about calculus is more important than the topics (which are pretty standardized) that you know. For example, when you look at the equation $x^3-x-1=0$ do you think purely algebraically? Or do you think in terms of a picture? Can you tie that picture to theorems, such as the intermediate value theorem?

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from a personal perspective, I never had any ability to do an $\epsilon - \delta$ proof for limits, similarly I never really understood the purpose of theorems either (so I never really remembered them). I only say these things because the answer seems a bit intimidating :P –  DanZimm May 15 '13 at 7:00
    
@DanZimm Oops - sorry, certainly not trying to be intimidating! But the question was basically "what parts of calculus are most helpful for continuation into real analysis". In that context, it's hard to avoid the fact that real analysis is the theory of calculus and, as such, gets heavily into proofs. –  Mark McClure May 15 '13 at 7:06
    
Yea that's true - just from a personal perspective, I remember knowing very little of the theory before my first analysis course –  DanZimm May 15 '13 at 7:19
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