Does $f(x + n)$ go to zero for almost all $x \in \mathbb{R}$ and $f$ integrable.

Question

Let $f \in \mathcal{L}^1(\mathbb{R})$. Is it true, that for almost all $x \in \mathbb{R}$ we have

$$\lim_{n \rightarrow \infty} f(x + n) = 0?$$

I was already able to prove that we have

$$\lim_{n \rightarrow \infty} \int_{[n, n+1]} f(x) \,d\mu(x) = 0$$

(using dominated convergence). How can I proceed from here?