Show that $\frac{1}{k}-\ln\left(\frac{k+1}{k}\right)$ is bounded by $\frac{1}{k^2}$

Question

I want to show that the following inequality holds for all $k\in\mathbb{N},k\geq 1$:

$$ \frac{1}{k}-\ln\left(\frac{k+1}{k}\right)\leq\frac{1}{k^2} $$

I already know that it is bounded. It also know that $1+x\leq e^x$ for all $x\in\mathbb{R}$. and that $e^x\leq\frac{1}{1-x}$ for all $x\in [0,1)$.