# Continuous limited function, thus uniformly continuous

1. Dec 10, 2008

1. The problem statement, all variables and given/known data
suppose f : [0,infinity) -> R is continuous, and there is an L in R, s.t. f(x) -> L, as x -> infinity. Prove that f is uniformly continuous on [0,infinity).

2. Relevant equations
limit at xo: |x-xo| < delta then |f(x) -L| < epsilon

continuous |x-x0| < delta then |f(x) - f(xo) | < epsilon

uniformly continuous: |x-y| < delta then |f(x) - f(y)| < epsilon.

3. The attempt at a solution
Let A be a number s.t.
if |x - y| < delta1 , |f(x) - L | < epsilon for all x in [a,infinity).
b/c f is continuous then |x-xo| < delta1 |f(x) - f(y) | < epsilon.
Thus f is uniformly continuous on [a,infinity).

Then,
b/c f is continuous it is uniformly continuous on any compact domain, namely [0,a].
|x-y| < delta2 ...

Then pick delta = min{delta1,delta2}, then the rest should follow.

Does that seem sound?