Let $f: \Bbb R \to \Bbb R$ be a continuous function satisfying $f(x+n) \to \infty$ as a sequence in $n$, for all $x$. Does $f$ satisfy $f(x) \to \infty$ as $x\to \infty$?
If we drop the continuity assumption then the claim is false, by considering a function tending more and more slowly to $\infty$ as we start at larger values in $(0,1)$. (Or many other examples)
As for context: a variant of this claim (when $f$ is analytic and we replace $n$ by a general increasing sequence $a_n$) could be useful to me at some technical exercise, and this is a simplification which I still cannot tackle.
Assuming by contradiction that $\exists M\forall x \exists x_0>x: f(x_0)\leqslant M$, we want to show that $\exists x \exists N>0 \forall n \exists n_0>n: f(x+n) \leqslant N$. No "quantifiers-level" logic seems to apply here.