TITLE: How do we know that $x^2 + \frac{1}{x^2}$ is greater or equal to $2$?
QUESTION [16 upvotes]: For one problem, we were supposed to know that: 
$$x^2 + \frac{1}{x^2}\geq 2.$$
How do you deduce this instantly when looking at the expression above?

REPLY [2 votes]: You said imediately.  So this argument is worse than the others but it was the first thing I thought of.
this is equivalent to showing if $w \ge 1$ then $w + 1/w \ge 2$ which is equivalent to showing $w-1 \le 1 - 1/w $.
And $1 - 1/w = (w -1)/w \le w-1$