Let $a \in \mathbb{R}$ and $f:[a,\infty) \rightarrow \mathbb{R}$ a continuous function. Suppose that there exists $L \in \mathbb{R}$ s.t. $\displaystyle\lim_{x\to\infty}f(x)=L$. Prove that $f$ is uniformly continuous on $[a,\infty)$.
I usually do fine however I am stuck at implementing my idea. I thought to assume the opposite*, then choose $\frac\varepsilon2$ at the limit definition where the epsilon is given by the assumption*. What do I miss?
Thanks