How to show that $a_n=\left(1+\frac{1}{n}\right)^n$ monotonic growing

Question

I have to show that

$a_n=\left(1+\frac{1}{n}\right)^n$

is monotonic increasing. I struggle alot and tryng to find a right form to epxress that,

$a_n\le a_{n+1}$

a little hint what might help would be great. Im in my first year in the introduction to real analysis, so tipps with derivative or more advanced arent usefull :/

I really only like to get a little hint :)

Answer 1

$$\frac{a_{n+1}}{a_n}=\frac{\left( 1+ \frac{1}{n+1} \right)^{n+1}}{\left( 1+ \frac{1}{n} \right)^{n}}=\frac{n+1}{n}\left( 1- \frac{1}{(n+1)^2} \right)^{n+1}>\\ >\frac{n+1}{n}\left( 1- \frac{1}{(n+1)} \right)=1$$

Answer 2

Let $$ f(x)=(1+\frac1x)^x $$ and show $f(x)$ is increasing.