algebra precalculus - Prove $left(1+frac1{n}right)^{n}

I am trying to prove the following by mathematical induction:
$$\left(1+\frac1{n}\right)^{n}<\left(1+\frac1{n+1}\right)^{n+1}$$
Other proofs without induction are found here: I have to show $(1+\frac1n)^n$ is monotonically increasing sequence.
But I am curious whether it can be proved by induction as well.

What I've tried so far:

The original inequality is equivalent to
$$(n+1)^{2n+1}So I have to show:

$$(n+2)^{2n+3}<(n+1)^{n+1}(n+3)^{n+2}$$
And,
$$(n+1)^{n+1}(n+3)^{n+2}=(n+1)\color{red}{n^n}\left(1+\frac1n\right)^n\cdot(n+3)\color{red}{(n+2)^{n+1}}\left(1+\frac1{n+2}\right)^{n+1}$$ $$>(n+1)(n+3)\cdot\color{red}{(n+1)^{2n+1}}\left(1+\frac1n\right)^n\left(1+\frac1{n+2}\right)^{n+1}$$ $$=(n+1)(n+3)\left(n+2+\frac1n\right)^n\left(n+1+\frac{n+1}{n+2}\right)^{n+1}$$
and I am stuck.