I'd love your help with this question:
Let $f$ be a continuous function in $[0,\infty)$, and $\lim_{x \to \infty }f(x)$ exist and finite.
I need to prove that $f(x)$ is bounded. Furthermore, if $\lim_{x \to \infty }f(x)=\lim_{x \to -\infty}f(x)$ and they are both finite, the function gets a maximum and minimum values.
What is the difference between being bounded and getting min and max? I know that under these conditions, this function is uniformly continuous.
Thank you