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 $\lim\limits_{n \to\infty}a_{n+1} = \lim\limits_{n\to\infty}a_n$?

Write $L=\lim_{n\to\infty}a_n$. This means that for every $\epsilon>0$ there exists a whole number $N$ such that for all integers $n\ge N$, $$|L-a_n|<\varepsilon$$ Now, to prove $\lim_{n\to\infty}a_{n+1}=L$ as well, consider any $n\ge N$ and note that $n+1\ge N$. So by our initial statement, $|L-a_{n+1}|<\varepsilon$. 
