<s>[INST] Prove that $2|ab| \leq a^2 + b^2$ and $|a|+|b| \leq \sqrt {2}(a^2+b^2)^{1/2}$ I am having issues solving these problems:
I tried using $(|a|+|b|)^2 \geq 0$ and $(|a|-|b|)^2 \geq 0$ but I am having problems constructing the proof. I need to gain some intuition on how to proceed.

Show that for $a,b\in (-\infty,+\infty)$,   the following inequalities apply:

a) $2|ab| \leq a^2 + b^2$
b) $|a|+|b| \leq \sqrt {2}(a^2+b^2)^{1/2}$
 [/INST] (a) $0 \leq (|a|-|b|)^2 = |a|^2 -2|a||b|+|b|^2 = a^2 - 2|ab|+ b^2\Rightarrow 2|ab| \leq a^2+b^2 $
(b) by (a) we have $(|a| + |b|)^2 = a^2 + 2|ab| + b^2 \leq a^2+b^2+a^2 +b^2 = 2(a^2+b^2) \Rightarrow |a|+|b| \leq \sqrt {2(a^2+b^2)}$
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