How can I prove the following inequality:
If $r>p$, then \begin{equation} \left\|X\right\|_{p}\le\left(\frac{r}{r-p}\right)^{1/p}\left\|X\right\| _{r,\infty}, \end{equation} where \begin{equation} \left\|X\right\|_p=\left(\operatorname{E}\left|X\right|^p\right)^{1/p}\qquad\text{and}\qquad\left\|X\right\| _{r,\infty}=\left(\sup _{t>0}\ t^r\Pr\left(\left|X\right|>t\right) \right)^{1/r}. \end{equation}
I found this inequality on page 10 of the book by Ledoux and Talagrand.
Thank you for your answers!