# Covergence in probability and almost surely

Let $X_n$ be a sequence of independent random variable which converges in probability to $X$. Prove $X$ is a constant.

Can someone give me a hint how I should go about proving this? I tried proving this by contradiction by saying $X$ taking 2 different values, but this can still still happen because $X$ is only almost surely constant.

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Hint 1: Passing to a subsequence, we can assume $X_n \to X$ almost surely.
Hint 2: Do you know the Kolmogorov zero-one law? $X$ is a tail random variable.