Prove $a^2 + b^2 \geq 2ab$ using Triangle Inequality Let $a$ and $b$ be two real, positive numbers. Is it possible to prove
$$a^2+b^2 \ge 2ab$$
using the Triangle Inequality?
This was suggested to me as a proof method but I have been unsuccessful so far.

By the triangle inequality $$|a-b|+|b-a|\geq|a-b+b-a|=0.$$
Thus, $$|a-b|\geq0$$ or
$$(a-b)^2\geq0$$ or
$$a^2+b^2\geq2ab.$$
