I'm trying to prove the following inequality, but don't have any idea on where to begin.
$$\inf_{k=0,1,2,\cdots} \frac{\mathbb{E}[|X|^k]}{\delta^k} \le \inf_{\lambda> 0} \frac{\mathbb{E}[e^{\lambda X}]}{e^{\lambda \delta}}$$
The RHS is the Chernoff bound on the tail probability distribution of $X$. The LHS is also another bound on the tail probability.