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?
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$\begingroup$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).
$\endgroup$ $\begingroup$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.
$\endgroup$ 5 $\begingroup$Analysis has evolved over three centuries. We need to take a different path with hindsight. One should introduce limit using neighborhoods rather than epsilon deltas. Also one can define phrases like sufficiently close , and selection of smallest neighborhood or largest n can be formalized in phrases. Further several example,s and finding limits in sequences series can be easily tackled using application of L hospital rule treating the variable as continuous. One needs to struggle if one does not use this rule and practices raw analysis We need to prove algebra of limits rigorously and here we need epsilons but not delta as we can use neighborhoods for independent variables. Rather than proving theorems on continuous functions it seems worthwhile to introduce metric space but as far as possible stick to neighborhoods. .
But one proves open ball is open set closed ball is closed set. arbitrary union and finite intersection.
One teaches both pathwise contentedness and contentedness but without stress on exotic examples and just to the point and in one of the basic proof, one models using bisection argument may be the theorem that interval is connected. One introduces compactness using Bolzano Weirstrass property. Proves it for R. One uses products and show product of complete spaces is complete.
Then one prove cantors theorem on complete spaces and from it derives Bolzano Weirstrass theorem in Euclidean spaces. Then one teaches continuous image of compact space is compact and then uniform convergence.nowhere one needs not so intuitive Heine Borel theorem Then comes to basic idea of derivative of one variable . One proves algebra of derivatives Draboux theeorem . But then one introduces Henstock-kurzweil integral and using it one derives both Lagranges mean value theorem and L hospitals rules as well as Taylors formula for one variable but vector mappiungs. One introduces notion of Banach space and Hilbert space Cauchy Schwartz inequality. Then one studies sequences and series tests for convergence in detail.
Admittedly one loses some sight in this synthesis
For example though contentedness and compactness are different, in case of real line they stem from one single notion of completeness. Yes we would need separate course on foundations simultaneously treating cantors theory and construction of real numbers.
I mean with lot of experience and hindsight one needs to approach real Analysis now in this fashion . In calculus one just avoids epsilon and deltas. Calculus is motivated geometrically
