Let $a\in\Bbb{R}$. Suppose $f$ is a real valued continuous function on $[a, \infty)$ satisfying that $\lim\limits_{x\to\infty}f(x)=L\in\Bbb{R}$ I need to show:
$f$ is bounded on $[a, \infty)$
Here is what I have so far: if $f$ is bounded on $[a, \infty)$, then there exists a constant $M$ s.t. $|f(x)| \le M$ for all $x \in [a, \infty)$
From the fact that $\lim\limits_{x\to\infty} f(x) = L$, I can say that given $\epsilon > 0$, there is a $\delta$ st $|f(x) - L| < \epsilon$.
If $L=0$, then $|f(x)| < \epsilon$. I can set $M \le \epsilon$, 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 would it work? Please provide me with some hints