How? If $\lim_{n \to +\infty} a_{n} = a$, then $\lim_{n \to +\infty} a_{n+m} = a$, where $m \in \mathbb{N}$

Question

I was watching a lecture on Mathematical Analysis and the teacher gave the following example :

$\lim_{n \to +\infty} a_{n} = a$, then $\lim_{n \to +\infty} a_{n+m} = a$, where $m \in \mathbb{N}$.

I got totally confused by this example and totally lost my understanding of the limits.

I need an explanation of this example. Is it correct? if so, then how?

Answer 1

$(a_n)$ converges to $a$ iff for all $\epsilon > 0$ there exists $N \geq 1$ such that $|a_n - a| < \epsilon$ for all $n \geq N$. If this is the case then for all $\varepsilon > 0$ there exists (you can take the same $N$, since $n+m > n \geq N$) $N \geq 1$ such that $|a_{n+m} - a| < \varepsilon$ for all $n \geq N$, as desired.