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.