Show that $\mathbf{E} a\xi 1_{\xi>a} \leq \sup_{t>0} t^2 \mathbf{P}(\xi>t)$ for $a>0$ and a positive random variable $\xi$. Show that $\mathbf{E} a\xi 1_{\xi>a} \leq \sup_{t>0} t^2 \mathbf{P}(\xi>t)$ for a positive random variable $\xi$. Here a>0 is arbitrary. 

For a non-negative random variable $X$, we have 
$$\mathbb E(X)=\int_0^{+\infty}\mathbb P\{X>t\}\mathrm dt,$$
hence 
$$\mathbb E(\xi\chi_{\{\xi>a\}})=a\mathbb P\{\xi>a\}+\int_a^{+\infty}\mathbb P\{\xi>t\}\mathrm dt.$$
Since $$\int_a^{+\infty}\mathbb P\{\xi>t\}\mathrm dt\leqslant \frac 1a\sup_tt^2\mathbb P\{X>t\},$$
we get the inequality (up to a factor $2$).
Since $a\xi\chi_{\{\xi>a\}}\geqslant a^2\chi_{\{\xi>a\}}$, we finally get by an integration that 
$$a^2\mathbb P\{\xi>a\}\leqslant \mathbb E(a\xi\chi_{\{\xi>a\}})\leqslant 2\sup_t t^2\mathbb P\{X>t\}.$$
