Bag of tricks in Advanced Calculus/ Real Analysis/Complex Analysis

Here are a couple of tricks and general plans of approach I know:

• If $x\in\Bbb R$ and for all $\epsilon>0$ we have that $|x|\leq\epsilon$, then $x=0$. I think of this fact as being Real Analysis in a nutshell.
• Never forget that if $A\subseteq\Bbb R$ is bounded above and $M=\sup A$, then for all $\epsilon>0$ there is a $y\in A$ such that $M< y+\epsilon$. Likewise if $A$ is bounded below and $m=\inf A$, then for all $\epsilon>0$ there is a $y\in A$ such that $y-\epsilon< m$. This is the most important tie between $\Bbb R$'s algebraic and ordering properties.
• Never underestimate the binomial theorem even if you just want an inequality. As an example, look at theorem 3.20(c) in Rudin and how he uses it.
• For all $x,y\in\Bbb R$ and any $\epsilon> 0$, we have the following inequality: $$|xy|\leq \frac{\epsilon\,x^2+\epsilon^{-1}\,y^2}{2}$$ This can be derived from the observiation that $(\epsilon\,|x|-|y|)^2\geq 0$. This inequality allows us to decide how much 'weight' we want to give to a particular term in a product. This can be used to show that the product of Riemann integrable functions is still Riemann integrable.
• A simple inequality to remember is $$(a+b)^p\leq 2^p(a^p+b^p)$$ for $a,b,p\geq 0$. This can be derived from the even simpler inequality $(a+b)\leq 2\max(a,b)$, again for positive values. This inequality can be used to show that the $L^p$ spaces are vector spaces.
• The Weierstrass M-test is the first friend you call when dealing with series of functions.
• Ask yourself whether the problem you're working on can be generalized to topology first. Think about compactness and connectedness and the abstract theorems about them you already know.
• Perhaps the greatest topological property that $\Bbb R$ has is second-countability. This means that $\Bbb R$ is hereditarily-separable, sequential, Frechet-Urysohn, and c.c.c. This property of $\Bbb R$ allows us to consider sequences and sequential continuity in place of neighborhoods and continuity. As an adage, if you are working with $\epsilon$, wonder to yourself if you can instead work with $1/n$ with $n\in\Bbb N$.

One `trick' that is used a lot in my analysis courses is: instead of showing that $x \leq y$ directly, it is usually a lot easier to show that, for all $\epsilon > 0:x \leq y + \epsilon$.

I'm not sure if you are requiring that a trick be something overly hard/creative or just something more along the lines of "Ahhh... I might not have thought of that, but now that I've seen it, I'd be able to do that again!", especially if it appears again and again.

If you mean the latter, keep in mind the ol' "add-and-subtract" or "$\varepsilon/3$ trick" where you insert a new term(s) that adds and then subtracts off some useful quantity, usually is followed by an appeal to the triangle inequality (or something similar) and some known estimates. A classic example is in proving that the uniform limit of continuous functions is continuous, where we use this to manufacture the terms leading to the $\varepsilon/3$'s.

It's not a complicated technique but certainly a recurring one in analysis.