**Lemma** Let $(a_{n})$ be a [[Convergence|sequence]]. For any fixed $k \in \mathbb{N}$, $\lim_{ n \to \infty } a_{n} = l \iff \lim_{ n \to \infty } a_{n+k} =l$ **Proof** Given $\epsilon>0$, choose $N$ such that $|a_{n}-l|<\epsilon$ for all $n \geq N$. Then if we take $n>N$, we have $n+k>n>N$ so $|a_{n+k} - l|<\epsilon$. Conversely given $\epsilon>0$, find $M$ such that for all $m \geq M,$ $|a_{m+k}-l|<\epsilon$. Then if we take $N=M+k$ we have $|a_{n}-l|<\epsilon$for all $n \geq N$. $\square$