Why does $\lim_{n \to\infty}a_{n+1} = \lim_{n\to\infty}a_n$? Assume that $\{a_n\}$ is a convergent sequence. How to use the definition of a limit of a sequence to prove that $\lim\limits_{n \to\infty}a_{n+1} = \lim\limits_{n\to\infty}a_n$?

I think this is just a matter of notation.
For $n \ge 1$,
considering $a_{n+1}$ means the sequence $a_2, a_3, a_4, a_5, \dots$, and
considering $a_n$ means the sequence $a_1, a_2, a_3, a_4, a_5, \dots$.
Limits are a matter of the eventual behaviour of the sequence, and it is not surprising that the limit (if it exists) is the same for "both" — it's the same sequence whose eventual behaviour you're considering.
Put differently, $$\lim_{n\to\infty} a_{n+1} = \lim_{m-1\to\infty} a_m = \lim_{m\to\infty} a_m = \lim_{n\to\infty} a_n$$
as $m-1\to\infty$ and $m\to\infty$ are the same thing.
