Lemma: for any

Proof: after simplifying the inequality becomes: for a fixed the quadratic function is convex. hence it attains its maximum at end

points, i.e. or now because also: because

now we may assume that let then:

since by Chebyshev's inequality we have: thus:

since we get: by theLemma. this proves the upper bound.

proving the lower bound is much easier: since applying the second part of Chebyshev's inequality to and this fact that whenever gives us: