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

my attempt by Daniel)

Nice attempt, Daniel! But presumably you could have started at line 6 of your solution, (c-a)^2 >= 0 which is true for all real numbers a and c.

Since the rest of the proof is still valid from that point on, perhaps, Pythagoras wasn't necessary!