TITLE: Show that $\left(1+\dfrac{1}{n}\right)^n$ is monotonically increasing
QUESTION [38 upvotes]: Show that $U_n:=\left(1+\dfrac{1}{n}\right)^n$, $n\in\Bbb N$, defines a monotonically increasing sequence.

I must show that $U_{n+1}-U_n\geq0$, i.e. $$\left(1+\dfrac{1}{n+1}\right)^{n+1}-\left(1+\dfrac{1}{n}\right)^n\geq0.$$
I am trying to go ahead of this step.

REPLY [4 votes]: My two cents (as later similar question closed):
$$\frac{a_{n+1}}{a_n}=\frac{\left( 1+ \frac{1}{n+1} \right)^{n+1}}{\left( 1+ \frac{1}{n} \right)^{n}}=\frac{n+1}{n}\left( 1- \frac{1}{(n+1)^2} \right)^{n+1}>\\
>\frac{n+1}{n}\left( 1- \frac{1}{(n+1)} \right)=1$$