I assume you're given that f is continuous?
Edit: Or you have to show existence of an such that f is bounded on [a,inf)? Use the definition of a limit at infinity, i.e. there is an X such that for x>X, |f(x)-L|<epsilon.
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)