TITLE: Prove existence
QUESTION [7 upvotes]: Let $f$ be a continuous bounded function on the interval $(a, +\infty)$ such that $\lim_{x \to+\infty}f(x)$ does not exist. Prove that for any $t \in\mathbb R$ there is a sequence $x_{n} \to +\infty$ that $f(x_{n} + t) = f(x_{n})$ for all $n \in\mathbb N$.

REPLY [10 votes]: Counterexample. Let $a=1,\ f(x)=\sin x+\frac1x$ for $x\in(1,+\infty)$, and $t=2\pi.$