Let

Prove directly from the definition of the limit that

for any constant .

*Proof.* Let be given. Since and we know there exist positive integers and such that

for all and all . Let . Then, for all we have

Hence,

*Proof.* Let be given. Since we know there exists a positive integer such that

Thus,

There is a small problem in the second proof: you cannot just divide by |c|, since it can be 0. Thus, there must be 2 cases: case |c| > 0, and c = 0.