Let be a real-valued function defined on the set of real numbers that satisfies

for all real numbers and . Prove that for all .

Proof:

Let be the assertion that Considering yields . This implies that ; denote this assertion by

Lemma 1: for all

Proof: Assume for the sake of contradiction that there exists a such that . Considering and letting we have that for all sufficiently small , Now considering and letting we have that which is a contradiction as desired.

Lemma 2: for all

Proof: Considering we have that so if then Combining this with Lemma 1 yields the desired result.

Lemma 3:

Proof: Considering and applying Lemma 2 immediately yields the desired result.

Lemma 4: for all

Proof: Considering immediately yields the desired result.

Now by Lemma 4 we have that and that so . Plugging this in to the statement of Lemma 2, we have the desired result.