If $\alpha _{n}\rightarrow \alpha$, then how does one show that for any j=1,2,... and $\epsilon> 0$, if $sup\int \left | x \right |^{j+\epsilon }d\alpha _{n}<\infty$, then $\int x^{j}d\alpha _{n}\rightarrow \int x^{j}d\alpha$ as $n \to \infty$?

What if $\epsilon= 0$? Is it stiil true? For example, if $sup\int x ^{2}d\alpha _{n}<\infty$, then is it necessarily the case that $\int x^{2}d\alpha _{n}\rightarrow \int x^{2}d\alpha$ as $n \to \infty$?