# Formally proving that $\lim\left[n/(n^2 + 1)\right] = 0$

Find the limit of $(a_n)$ where $a_n = n/(n^2 + 1)$ (formally, using the definition of the limit).

Clearly, the limit is $0$; however, I'm having trouble finishing the proof. Given any $\epsilon > 0$, I know that I need to find a $N \in \mathbb{R}$ such that $\forall n \geq N$, $\left| a_n \right| < \epsilon$. I'm unsure of how exactly to proceed.

Hint: Start by writing $$0\leq\frac{n}{n^2+1}=\frac{1}{n+\frac{1}{n}}\leq\frac{1}{n}.$$ Does this show you how to choose $N\in\mathbb{N}$ so that $n\geq N$ implies $\lvert a_n\rvert<\epsilon$?