\begin{align*} \mathbb{E}\left|X_{n}\right| & =\int_{\left\{ \left|X_{n}\right|\le1\right\} }\left|X_{n}\right|d\mathbb{P}+\int_{\left\{ \left|X_{n}\right|>1\right\} }\left|X_{n}\right|d\mathbb{P}\ & \le\mathbb{P}\left\{ \left|X_{n}\right|\le1\right\} +\int_{\left\{ \left|X_{n}\right|>1\right\} }\left|X_{n}\right|\log\left(1+\left|X_{n}\right|\right)d\mathbb{P}\ & \le1+\mathbb{E}\left|X_{n}\right|\log\left(1+\left|X_{n}\right|\right)<\infty. \end{align*}
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