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

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?

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.