I came across the following problem which I would like to solve.
Let $f: \mathbb R \rightarrow \mathbb R$ be a lebesgue integrable function. Prove the following:
$$\lim_{n\rightarrow\infty}f(x+n)=\lim_{n\rightarrow\infty}f(x-n)=0$$
Would like to solve this myself (partially) and would appreciate (only) hints
Thanks
Integrability is not enough. Consider $$\varphi(x) = \sum\limits_{k = 0}^\infty k \, \chi_{[k - 2^{-k}, \, k + 2^{-k}]}(x),$$
where $$ \chi_A (x) = \begin{cases} x & \text{if} \; x \in A , \\ 0, & \text{otherwise}; \end{cases}$$
Then $$ \int \varphi(x) \, d\mu(x) = \sum\limits_{k = 0}^{\infty} k \mu[k-2^{-k}, \, k + 2^{-k}] = \sum\limits_{k = 0}^\infty 2k 2^{-k} < \infty, $$
so $\varphi$ is integrable. But $$\lim\limits_{k \to \infty, \, k \in \mathbb{N}} \varphi(k) = \lim\limits_{k \to \infty, \, k \in \mathbb{N}} k = \infty.$$