Prove Using Definition that the Sequence $\frac{n+1}{\sqrt{n}+2}$ Diverges to Infinity Formally, a sequence $x_n$ diverges to infinity whenever for all $M>0$ there exists $N(M)$ such that $n>N(M)$ implies $x_n>M$.
Prove formally, using the definition, that the following sequence diverges to infinity:
\begin{equation}
x_n=\frac{n+1}{\sqrt{n}+2}
\end{equation}
Proof. Let $M>0$. Take $N(M)=?$...

Hint You want for all $n >N(M)$ to have:
$$\frac{n+1}{\sqrt{n}+2} >M $$
Solve this inequality for $n$.
Or easier, you can observe that
$$\frac{n+1}{\sqrt{n}+2}> \frac{n-4}{\sqrt{n}+2}=\frac{(\sqrt{n}-2)(\sqrt{n}+2)}{\sqrt{n}+2}$$
and if you make this last term greater than $M$ you are done.
