Prove that the sequence $\left\{\frac{n^2+1}{n+2}\right\}_{n=1}^\infty$ diverges to $\infty$ How would I be able to show that the sequence $\left\{\frac{n^2+1}{n+2}\right\}_{n=1}^\infty$ diverges to $\infty$?
Here is what I do know: Suppose $M>0$ is given. We must find $N \in \mathrm N$ such that $n \ge N \Rightarrow a_n = \frac{n^2+1}{n+2} > M$. So, we may observe that for any $n \ge 1$...?
And right there is where I get stuck. Can anyone help and explain the next part to me? Thank you in advance.

If $n>1$, $$\frac{n^2+1}{n+2}\geqslant \frac{n^2+1}{2n}\geqslant \frac{n^2}{2n}\geqslant \frac n2\to\infty$$
