Finiteness of the supremum of a function

Question

Suppose that $\{x_n\}$ is a sequence in $\mathbb{R}$ s.t. $\sup_n|x_n|<\infty$ and let $f$ be a continuous function on $\mathbb{R}$. Is it true that $$ \sup_n|f(x_n)|<\infty? $$

Answer 1

Yes. Since $\sup\{|x_n|\}$ is bounded, the sequence $|x_n|$ lies in some closed interval, $[a,b]$. By the extreme value theorem, $f$ is bounded on $[a,b]$, hence $$\sup_n |f(x_n)| \leq \sup_{[a,b]}|f(x)| < \infty.$$

Answer 2

Yes, because the condition imposes that the sequence $x_n$ is bounded, let's say bounded by $M$. Moreover, by continuity we have that $ f$ is bounded on any compact, i.e. closed and bounded, interval. In particular, it is bounded on $[-M,M]$. Thus the supremum is also bounded.