**Lemma** Let $(a_{n})$ be a [[Convergence|sequence]] of integers $(a_{n}\in \mathbb{Z}$ for every $n$) and assume that $a_{n} \to L$ as $n \to \infty$ for some $L \in \mathbb{R}$. Then $L \in \mathbb{Z}$ and $(a_{n})$ must eventually by constant. **Proof** Taking $\epsilon = \frac{1}{2}$ in definition of convergence, there exists $N$ such that for all $n \geq N$ we have $|a_{n}-L|< \frac{1}{2}$. Now if we take $n \geq N$ we have $|a_{n}-a_{N}|\leq |a_{n}-L|+|L-a_{N}| <1$Since both $a_{n}$ and $a_{N}$ are integers, we conclude that $a_{n}=a_{N}$ for all $n \geq N$. In other words, the sequence is eventually constant, so the limit $L$ equals $a_{N}$ and in particular is an integer.