Showing $\{X_n\}$ is uniformly integrable

Question

so I want to prove that $\{X_n\}$ is uniformly integrable, given that E[$|X_n|^2$] < C for every n where C is a constant.

so using the markov inequality we get P($|X_n|^2$ $\geq$ N) $\leq$ $\frac{(E[|X_n|^2])}{N}$ $\leq$ $\frac{C}{N}$ , $\lim \limits_{N \to \infty}$$\frac{C}{N}$ = 0 therefore we can conclude that:$\lim _{N\rightarrow \infty}\int_N^\infty P(|X_n|^2\geq N)dx \rightarrow 0$ but then I get that $\{X_n^2\}$ is uniformly integrable how can I continiue from here or did I missed something?

Answer 1

Note that $$1\{|X_n| \geq N\} N |X_n| \leq 1\{|X_n| \geq N\} |X_n|^2,$$ hence $$\int_{|X_n| \geq N} |X_n| dP\leq \frac{1}{N} \int_{|X_n| \geq N} |X_n|^2 dP.$$ Can you conclude?