Infinite series proof

Question

I need to prove this statement using the limit definition.:

$ a_n$ converges $\Rightarrow a_{n+1} converges$ to the same limit.

I started with $\epsilon>0$ $\exists n_0$ $\forall n>n_0$, $|a_n -L|<\epsilon$

What is the right way to finish this proof?

Thank you.

Answer 1

Take $N_0=n_0-1$. Then\begin{align}n>N_0&\iff n>n_0-1\\&\iff n+1>n_0\\&\implies|a_{n+1}-L|<\varepsilon.\end{align}