Let $\ (\Omega,F,P)$ a probability space $X, X_n, n=1,2,\ldots$ are real valued random variables on $ (\Omega,F,P)$. Assume that $\ E[e^{c|X|}]< \infty$ for some $c>0$. Define $\ X_n = n(e^{X/n}-1), n\geq 1 $. By MVT for every $\ n\geq 1$ and every $\omega \in \Omega $ there exists $\ t_n(\omega) \in (0,1/n)$ s.t $\ X_n(\omega)$= $\ X(\omega)$ $\ e^{t_n(\omega)} $. $\ X(\omega)$ choose $\ n_0$ s.t $c>2/n_0$.
Find an integrable random variable $Y$ on $\ (\Omega,F,P)$ s.t $|X_n| \leq Y $ for all $\ n\geq n_0$