Example 1.30 from Introduction to Mathematical Thinking by John D'Angelo

Question

I understand the example. However, I am curious how the author knows to start the argument with $$(1-\lvert x\rvert)^2\ge0.$$ What algorithm does he follow that begins with this assumption?

Answer 1

A way to "guess" it would be to first note that the function is odd i.e. $\,f(-x)=-f(x)\,$, and therefore it is enough to prove the inequality for $\,x \ge 0\,$. But in that case $\,|x| = x \ge 0\,$, so:

$$ \dfrac{x}{1+x^2} \le \dfrac{1}{2} \;\;\iff\;\; 2x \le 1 + x^2 \;\;\iff\;\; 0 \le 1-2x+x^2 = (1-x)^2 $$

Now that you have the "clue" figured out, dress it up as $\,\left(1-|x|\right)^2\,$ to account for negative $x$'s, then plug it all the way back into the original statement, without any hint whatsoever of how you got it, and it suddenly looks like a bit of black magic ;-)

Answer 2

Mathematical thinking doesn't always come from an algorithm. Also, you shouldn't assume that just because the author wrote $(1-|x|)^2 \geq 0$, it was the first thing that popped into his head. When you write a proof, often it comes together in bits and pieces. You might start at the beginning and see where your guesses take you, or you might jump to the end and see what it takes to get there.

But here's a rule of thumb which the author may be relying on: it's been said that all inequalities can be derived from the fact that for all real $a$, $a^2 \geq 0$. The author has used this fact, substituting $a = 1-|x|$.