What are some useful problem solving strategies for real analysis?

Question

In this blog, Professor Tao exhibited some problem solving strategies that can help students in their study of (mostly) measure theory and some are intended for analysis in general.

I'd love to see some other good "tricks" that students of real analysis would like to learn and master since they'd make their life easier in proving theorems and doing exercises.

A more preferable answer would be one that includes a "trick" or a "strategy" with an example where this trick is already useful (for example, in proving such-and-such theorm)

Answer 1

The two volumes 'Solving Problems in Mathematical Analysis by Tomasz Radożycki' (https://link.springer.com/book/10.1007/978-3-030-35844-0) contain some very good problem-solving strategies for real analysis.

Answer 2

Look for monotone sequences and functions

The real numbers have the monotone convergence theorem: every bounded, monotonic sequence in $\mathbb R$ has a limit in $\mathbb R$. The rational numbers do not have this theorem: there are monotonic, bounded sequences in $\mathbb Q$ that do not have a limit in $\mathbb Q$. An example of such a sequence can be given by defining $\{x_n\}_{n=0}^\infty$ recursively via the equations

$$x_{n+1}=\frac{x_n}{2}+\frac{1}{x_n},x_0=2$$

This sequence consists entirely of rational numbers, is monotone decreasing, and bounded between $0$ and $2$, but the only possible limit of the sequence would have to be a positive square root of $2$. No such number exists in $\mathbb Q$.

In fact, up to isomorphism, $\mathbb R$ is the only ordered field that has the monotone convergence theorem. Thus, it should make sense that the proof of any theorem that is true in $\mathbb R$ but not in other ordered fields (e.g. $\mathbb Q$) depends almost exclusively on this property holding in $\mathbb R$.

This theorem implies another important statement and its many cousins: for any bounded, monotonic function defined on an interval $(a,\infty)$ ($a\in\mathbb R$), $\lim_{x\to\infty}f(x)$ exists. All these statements on monotone sequences and functions are equally important and useful.

When should we use these theorems?

The monotone convergence theorem and its counterpart for functions establish the existence of numbers, but are only useful if one is able to construct monotonic sequences and functions. For an instructive example, let's prove that any absolutely convergent improper integral also converges in the standard sense:

Suppose $f:[0,\infty)\to\mathbb R$ is integrable on any closed interval $[a,b]\subset[0,\infty)$; this allows us to make sense of the expression $\int_0^t f(x)dx$ for any $t>0$. If $\int_0^\infty |f(x)|dx$ converges, then $\int_0^\infty f(x)dx$ converges.

Proof: if $f$ is integrable on any closed interval $[a,b]\subset[0,\infty)$, then so is $|f|$ (proof omitted), so we can make sense of the expression $\int_0^t |f(x)|dx$ for any $t>0$ and hence speak of the improper integral $\int_0^\infty |f(x)|dx$ and its convergence.

Define the functions $f^+:[0,\infty)\to\mathbb R$ and $f^-:[0,\infty)\to\mathbb R$ by

$$f^+ (t)=\begin{cases} f(t) &\text{ if } f(t)\geq 0\\ 0 &\text{ if } f(t)< 0\end{cases}$$ $$\text{and}$$ $$f^- (t)=\begin{cases} f(t) &\text{ if } f(t)\leq 0\\ 0 &\text{ if } f(t)> 0\end{cases}$$

These are known as the positive and negative parts of $f$, respectively. An immediate consequence of these definitions are the identities $f =f^+ + f^-$ and $|f|=f^+ - f^-$. Standard theorems from integral calculus also show that $f^+$ and $f^-$ are integrable.

Clearly $f^+$ is non-negative and $f^-$ is non-positive, so the integrals

$$t\mapsto\int_0^t f^+(x) dx\text{ and }t\mapsto\int_0^t f^-(x) dx$$

are monotone increasing and monotone decreasing, respectively. Since the identity $|f|=f^+ - f^-$ implies that for any fixed $t>0$,

\begin{align*} \int_0^t f^+ (x) dx &= \int_0^t \left[|f(x)| + f^- (x)\right]dx\\ &= \int_0^t |f(x)|dx + \int_0^t f^- (x)dx\\ &\leq \int_0^\infty |f(x)|dx + 0 \text{ (since }f^-(x)\leq 0\text{ and }|f(x)|\geq 0\text{)}\\ &= \int_0^\infty |f(x)|dx \end{align*}

and

\begin{align*} \int_0^t f^- (x) dx &= \int_0^t \left[f^+ (x)-|f(x)|\right]dx\\ &= \int_0^t f^+(x)dx - \int_0^t |f(x)|dx\\ &\geq 0 - \int_0^\infty |f(x)|dx \text{ (since }f^+(x)\geq 0\text{)}\\ &= -\int_0^\infty |f(x)|dx \end{align*}

we also see that the integrals $\int_0^t f^+ (x) dx$ and $\int_0^t f^- (x) dx$ are bounded above and below, respectively. It follows that

$$\lim_{t\to\infty}\int_0^t f^+(x) dx\text{ and } \lim_{t\to\infty}\int_0^t f^-(x) dx$$

both exist, and hence so does

\begin{align*} \lim_{t\to\infty}\int_0^t f(x) dx &= \lim_{t\to\infty}\int_0^t \left(f^+(x)+ f^-(x)\right)dx\\ &= \lim_{t\to\infty}\left(\int_0^t f^+(x) dx+\int_0^t f^-(x) dx\right) \end{align*}

Some Intuition for the Ideas in this Argument

Geometrically, when we stick absolute value brackets around $f$, the negative portions of $f$ that were originally below the $x$-axis flip about this axis to be mirrored on top of it. The convergence of $\int_0^\infty |f(x)|dx$ is the statement that the area between the $x$-axis and $|f(x)|$ is finite. Since this area is made up of the areas of the positive regions and negative regions of $f$, it should follow that these areas are also finite. But then the signed sum of these areas is also finite, and since this area is $\int_0^\infty f(x) dx$, this integral should converge.

Integrating the functions $f^+$ and $f^-$ captures the positive areas and negative areas between $f$ and the $x$-axis, respectively. This is one of the reasons for introducing them. The other reason is that integrating $f^+$ and $f^-$ produces monotone functions. We want this because monotone functions are precisely the objects of interest to our monotone convergence theorems. All that remains is to show the resulting integrals are bounded.

Answer 3

I like the book M.Steele, The Cauchy--Schwarz master class, Cambridge university press. It takes the basic Cauchy--Schwarz inequality as a testing ground to demonstrate a wealth of strategies, also beyond analysis.

Another one somewhat in a similar spirit is S.Mahajan, Street-fighting mathematics, MIT press. This is really "a bag of tricks". All of those are very nice and I learned much there as a student.