Let $f:(0,+\infty)\to\mathbb{R}$ be continuous and bounded. Let $\xi>0$. Show that there is a sequence $(x_n)$ in $(0,+\infty)$ with $x_n\to\infty$ s.t.
$$\lim_{n\to\infty}|f(x_n+\xi)-f(x_n)|=0.$$
I tried to prove by contradiction. Assume this was false, then for any sequence $(x_n)$ with $x_n\to\infty$ and for any $n\in\mathbb{N}$, there exists $\varepsilon>0$ and $n_0\geqslant n$ s.t.
$$|f(x_{n_0}+\xi)-f(x_{n_0})|\geqslant\varepsilon$$
I want to conclude that in this case, $f$ cannot be bounded. I was stuck here. Am I on the right track? Can we prove this directly? I mean, without proof by contradiction.