How does one get better at real analysis proofs?

Question

How does one proceed through a math proof in real analysis? My instructor always says make a diagram, but I am not a visual learner. It seems that whenever I write out the definition of an assumption, then I cannot make the next logical step. Also, when I go to try to verify that my proof is correct, I ask myself questions like, "why must this be true?" but the proof does not end up not being air tight. For those that have had real analysis, what did you do to master proofs and do the exercises?

Answer 1

Most of the theorems in real-analysis (especially those in introductory chapters) are intuitive and based on the concept of inequalities. If one understands the concept of inequalities (not in the sense of memorizing AM greater than equal to GM or other famous inequalities) in terms of comparison of numbers most of the common proofs are trivial applications of the definitions.

I will provide two examples:

1) If $f$ is continuous at $x = a$ and $f(a)$ is positive then there is a neighborhood of $a$ in which $f$ is positive.

Now one has to know what is meant by continuity to prove this. Informally this means that values of $f(x)$ are arbitrarily near $f(a)$ if $x$ is sufficiently near $a$. The $\epsilon, \delta$ are used to quantify "arbitrarily" and "sufficiently" in a formal manner. Now if we see that $f(a)$ is positive then there is a range of values near $f(a)$ which are positive. Hence if $x$ is sufficiently close to $a$, $f(x)$ will take values in the range near $f(a)$ and these are all positive as mentioned in last sentence.

2) If $f(x) \leq g(x)$ in a neighborhood of $a$ and both $\lim_{x \to a}f(x), \lim_{x \to a}g(x)$ exist then $\lim_{x \to a}f(x) \leq \lim_{x \to a}g(x)$.

Clearly let $A = \lim_{x \to a}f(x), B = \lim_{x \to a}g(x)$. Suppose $A > B$. Now values of $f(x)$ are near $A$ when $x$ is near $a$. Similarly values of $g(x)$ are near $B$. Since $A > B$ we can obviously make values of $f$ much nearer to $A$ compared to $B$ and values of $g$ much nearer to $B$ compared to $A$. We will find that this leads to values of $f$ being greater than some values of $g$ and we get contradiction.

Thinking in terms of inequalities as a way of comparing magnitudes and numbers is the key to these kinds of proofs. However thinking in this fashion is not easy for a beginner as he is trained to think in terms of operations like $+, -, \times, /$ and not $< , >$. As a further example consider the two following facts:

a) There is no rational number whose square is equal to $2$ (i.e. $\sqrt{2}$ is an irrational number).

b) If $a$ is a positive rational such that $a^{2} < 2$ then there exists another rational $b$ such that $a < b$ and $b^{2} < 2$.

The proof of statement a) is mostly algebraical and can be figured out easily if we know simple facts about integers and their factorization. The proof of statement b) is not easy unless we know how to deal with inequalities (reader can convince himself by trying to prove this). I consider this to be the fundamental difference between algebraic and analytic approaches and a beginner in analysis must make a transition from understanding statements like a) to understanding statements like b).

Answer 2

I think the best thing to do is to learn and understand the proofs of the theorem you do in class. The key thing about analysis (as opposed to algebra) is that all the proofs have a pattern to them.

For example: The proof of sequence of continuous functions converges uniformly to a continuous function uses the idea called $\frac{\epsilon}{3}$ argument. By knowing this technique, you can do a number of other proof that involves convergence and continuity.