Originally Posted by

**ThePerfectHacker** Define

and show

. Now

. If there was

such that

then since

satisfies Rolle's theorem on

it would mean there is a point

such that

and this is a contradiction. Thus,

for

. If there was

such that

then since

satisfies IVT it means there would be

such that

. But this is a contradiction because

is increasing on

. Thus,

.

well, if we prove that for , then we're done because that means is increasing and thus:

the inequality is clearly true for x = 0. so we assume that x > 0. so the claim is:

suppose for now that a > 1. since we have thus: hence which obviously holds

for a = 1 as well. therefore:

which proves (1).

Q.E.D.