The basic arithmetic mean-geometric mean (AM-GM) inequality states simply that if x and y
are nonnegative real numbers, then.
(x+y)/2 ≥ √(x*y), with equality if and
only if x = y.

There are various proofs for this theorem (for any number of values), inter alia Polya, Cauchy, by induction etc.

Now derive your proof directly from Pythagoras' formula a^{2}+b^{2} = c^{2}, a ≠ b.

(In reply to

Solution by Bractals)

Bravo!Excellent work done thats a very nice way to prove the inequality.

I thank you very much for the solution you gave.