Convergence in probability of sum of bounded random variables implies finite expectation

Question

Let $(X_i)$ be sequence of independent random variables with $|X_i| < 1$ almost-surely and define $M_n :=\sum_{i=1}^{n}X_i$. Suppose it is known that $M_n$ converges in probability to some random variable $M$, where $M < \infty$ almost-surely. Does it follow that $\sum_{i=1}^{\infty}\mathbb{E}[X_i] < \infty$?