Why is this inequality true? $(a+b)^2\leq 2(a^2+b^2)$ Why is this inequality true? $a,b$ are real numbers.
$$
(a+b)^2=a^2+2ab+b^2\leq 2(a^2+b^2)
$$
I know $(a+b)^2=a^2+2ab+b^2 \geq 0$, but then?

$$(a-b)^2\geq0\\a^2-2ab+b^2\geq0\\a^2+b^2\geq2ab\\2a^2+2b^2\geq a^2+2ab+b^2\\2(a^2+b^2)\geq(a+b)^2$$
