Is proving LHS - RHS $\gt 0$ a valid method of proof?

Question

In a question such as

Prove $(a^2+b^2)/2 \gt [(a+b)/2]^2$

is proving that LHS - RHS $\gt 0$ a valid method of proof? (Eventually getting $[(a-b)^2]/4 \gt 0$)

Would I be better off using proof by contradiction?

Thanks in advance.

Answer 1

In this particular case and in most other cases, the best strategy to prove an inequality of type $$F(x,y) \geq G(x,y)$$ is to perform operations on it to obtain an equivalent inequality of the form $$H(x,y)^2 \geq 0$$ or other well-known inequalities such $AM\geq GM$ (link), Cauchy-Schwarz inequality (link) and others.