calculus - Prove by Mean Value Theorem $frac{x}{1+x}

Prove for $x>0$
$$
\frac{x}{1+x}<\ln(1+x)
$$
I tried writing $\ln(1+x)=\ln(1+x)-\ln(1)$ and using the MVT for the $(1,1+x)$ interval. I eventually could prove the inequality but how do I have to prove even for $(0,1)$