Proving an inequality involving the logarithm function: $frac{1}{n+1} leq ln left(1+frac 1nright) leq frac 1n$

The question is to prove the inequality

$$\frac{1}{n+1} \leq \ln \left(1+\frac 1n\right) \leq \frac 1n\\\forall n \geq 1, n\in \mathbb N$$

I tried using Taylor expansion but couldn't figure out anything. Any ideas? Thanks.

Answer

Suppose $n

$$\frac{1}{n+1}<\frac{1}{x}<\frac{1}{n}.$$
Integrate this with respect to $x$, from $n$ to $n+1$. Then

$$\frac{1}{n+1}<\log{(n+1)}-\log{n}<\frac{1}{n},$$
and the middle is $\log{(1+\frac{1}{n})}$.