# Some limit problem in a light-tailed Compound Poisson driven queue

As part of a homework exercise on a light-tailed Compound Poisson driven queue (i.e. a Compound Poisson process reflected at 0) I need to show that when $$B$$ denotes the job size of the Compound Poisson process, under the assumption that the mean of $$B$$ is finite we have that $$\lim_{y\rightarrow \infty}e^{y}y\mathbb{P}(B > y) = 0.$$ It seems to me that either the finiteness of the mean, or $$X$$ being light-tailed (i.e. $$\mathbb{E}e^{\omega X_1} = 1$$ for some $$\omega >0$$ and $$\mathbb{E}(X_1e^{\omega X_1}) < \infty$$) must imply that the probability $$\mathbb{P}(B > y)$$ must go to 0 superexponentially. However, I don't know how to show this.