analysis - Proving an inequality without an integral: $frac {1}{x+1}leq ln (1+x)- ln (x) leq frac {1}{x}$

I would like to prove the following inequality without integration; could you help?

$$\frac {1}{x+1}\leq \ln (1+x)- \ln (x) \leq \frac {1}{x}, \quad x > 0.$$

I can however differentiate this.

Thanks in advance.