uniformly continuous and boundeness

Let a be in R. Suppose f is a real valued function on [a, infinity) satisfying that lim as x -> infinity of f(x) is L, where L is in R.

I need to show:

f is bounded on [a, infinity)

here is what I have so far:

if f is bounded on [a, infinity), then there exists a constant M s.t. |f(x)| <= M for all x in [a, infinity)

From the fact that lim f(x) = L, I can say that given epsilon > 0, there is a delta s.t |f(x) - L| < epsilon.

if L=0, then |f(x)| < epsilon. I can set M <= to epsilon, but I don't think this last part makes any sense because there is no way I can claim that L =0

or what if I say that since |f(x) - L| < epsilon, then |f(x) - L + L| < epsilon + |L|. If I let M = epsilon + |L|, then it would work, wouldn't it?

Please provide me with some hints

By the way, the second part of the problem is to show f is uniformly continuous on [a, infinity)