How to show that a continuous function is bounded on an interval

Question

For a continuous function $f :[0, \infty) \rightarrow [0, \infty)$, $(x_n)$ is a sequence of positive real numbers diverging to infinity. If $f(x_n) \rightarrow 0$, is $f$ bounded on $[0,\infty)$?

Are there any theorems that can help in proving/disproving whether $f$ is bounded on $[0, \infty)$ or not?

Answer 1

No, take $f(x) = |x\sin(x)|$ and $x_n = 2\pi n$

But, if the statement is true for every sequence then $f$ is bounded with limit $0$ at infinity.