Subsequences of a convergent sequence converge to the same limit

Statement

Let $(a_n)$ be a real sequence that converges to a real number $L$. Let $(a_{n_k})_{k \in \mathbb{N}}$ be a subsequence of $(a_n)$. Then $a_{n_k} \to L$ as $k \to \infty$.

Remarks

For the proof, let $\varepsilon > 0$ and take $N$ with $|a_n - L| < \varepsilon$ for $n \ge N$. Since $n_k \ge k$, we get $|a_{n_k} - L| < \varepsilon$ for all $k \ge N$. It gives a test for divergence. If two subsequences have different limits, the sequence diverges. For $(-1)^n$ the even terms tend to $1$ and the odd terms tend to $-1$.

Explanations

L1 human

If all terms eventually stay close to $L$, then the terms that you keep also stay close to $L$. Deleting terms cannot change where the sequence is going.