Suppose $\{x_n\}_{n \geq 1}$ is a real valued sequence such that for every $a, r \in \mathbb Z_+$, we have that

$$\lim_{N \to \infty} \frac{1}{N} \sum_{i = 0}^{N-1} x_{a + ir}$$ exists and equals $L$ for some $L \in \mathbb R$.

**Question:** Is it true that $x_n \to L$ along a subsequence of natural density $1$? That is, does there exist a subsequence $n_k$ with

$$\lim_{M \to \infty} \frac{|\{k \in \mathbb N \, | \, n_k \leq M\}|}{M} = 1$$

such that $x_{n_k} \to L$?