This question reminded me of a better one: Prove that
Spoiler:
I'll prove what's in the spoiler box (the stronger result):
We need to be careful about in the line above. This is just one value though, so check it manually and assume .
Now let's find the maximum of .
so solving we see .
But a little work shows which proves the claim.
p.s. I'm sure there's a much more elegant proof.