If $f:[a,\infty) \to \mathbb R$ is uniformly continuous and $\lim_{x \to \infty}f(x)$ is finite, then show that $f$ is bounded on $[a,\infty)$
I don't know if I am completely right or not.
Help me with this or correct me if find me wrong somewhere.
Let $ \lim_{x \to \infty} f(x)$ finite, say $l$.
For a given $\epsilon>0$, $\exists$ a real number $b$ such that
$x>b$ $\implies$ $l-\epsilon<f(x)<l+\epsilon$.
Also,Uniform continuity of $f(x)$ implies continuity. Hence, $f(x)$ is continuous on $[a,b+1]$.
Every continuous function on a closed and bounded interval is bounded $\implies \exists$ a real number say, $m$ such that $|f(x)|<m$ $\forall x \in [a,b+1]$.
Now, let $p =\max\{l-\epsilon , l+\epsilon , m\}$.
Hence, $|f(x)|<p$ $\forall x \in [a,\infty)$.