Bag of tricks in Advanced Calculus/ Real Analysis/Complex Analysis I am studying for an exam and I have been studying my butt off
during the winter break for it. During the course of my study I have written
down quite a number of tricks, which in my opinion were 'outrageous' :-). Meaning there
was no way I would come up with that during an exam if I hadn't seen that before.
Couple of examples.


*

*Sometimes, when you want to prove something about $\max$, $\min$, you write ( I got this from Baby Rudin)


$$ \max(a,b)=\frac{a+b+ \vert a-b \vert} {2} $$
$$ \min (a,b)= \frac{a+b-|a-b|} {2} $$


*

*To prove Hölder's inequality (in its simplest case)
You write $\int (f+tg)^2 \geq 0$ and since this stays positive
you get that the discriminant of this must be negative, and magically
you get your Hölder inequality.

*When you want to show something about distinct zeroes of complex functions you kind of eliminate the zeroes of f by dividing them with the appropriate Möbius transforms and you still get an analytic functions which has nice properties.
The value of these is that they can be used in other contexts to write neat proofs. 
That's what I mean by "tricks". This might be difficult to answer,
but what are some of the tricks you wise folks have up your sleeve when it comes to
Advanced Calculus (Both single variable, multivariable) and complex Analysis.
Anything you have to share will be greatly appreciated. Thanks so much for all your help.

Now hear this. Suppose that $1/p + 1/q = 1$.  Then if you exploit the convexity of the log function you can show that for $x, y \ge 0$
$$xy \le {x^p\over p} + {x^q\over q}.$$
This is pivotal in proving Hölder's inequality.
