MHB Is it Possible for a Random Variable to Have a Variance of Zero?

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Here is this week's POTW:

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Let $X$ be random variable whose variance is zero. Prove that with probability one, $X = \Bbb E[X]$.

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No one answered this week's problem correctly. Here is my solution:

Spoiler

For convenience, let $\mu = \Bbb E[X]$. The sequence of events $(|X - \mu| > 1/n)$, $n\in \Bbb N$, increases to the event $(X \neq \mu)$, so $P(X\neq \mu) = \lim\limits_{n\to \infty} P(|X - \mu| > 1/n)$. By Markov's inequality and the assumption $\operatorname{Var}(X) = 0$, we have $$P(|X - \mu| > 1/n) = P(|X - \mu|^2 > 1/n^2) \le n^2 \operatorname{Var}(X) = 0$$ for all $n\in \Bbb N$. Therefore, $P(X\neq \mu) = 0$. In other words, $X = \mu$ with probability one.