Let $f: [0,\infty) →\mathbb{R}$ uniformly continuous such that $\lim_{n \to +\infty} f(x+n)=0\; \forall x\geq 0$: show that $f(x) =0$ as $x\to\infty$

Question

To me it seems clear since its valid for any $x \geq 0$ so just take any $x$ greater than an given $n$ for which $|f(n)| < \varepsilon$ for some $n > N$. I can't see why uniform continuity is needed in this case.

Feel free to give me material or hints, thanks.