Undergrad Convergence in Probability

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Prove that if $\{X_n\}_{n = 1}^\infty$ is a sequence of real random variables on probability space $(\Omega, \mathscr{F},\mathbb{P})$ such that $\lim_n \mathbb{E}[X_n] = \mu$ and $\lim_n \operatorname{Var}[X_n] = 0$, then $X_n$ converges to $\mu$ in probability.

 

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Spoiler

We're going to use
https://en.m.wikipedia.org/wiki/Chebyshev's_inequality

For any $m$, there exists $N$ such that for $n>N$, we have $P(|X_n-\mu | > 1/m) < 1/m^2$, applying Chebyshev's inequality with $\sigma < 1/m^3$ and $k=m$, and $N$ picked such that $var(X_n) < 1/m^3$ for $n>N$.

That's pretty much it, since $m$ it's arbitrary.

For any $\epsilon >0$, for any $m$ such that $1/m<\epsilon$, and for $n$ large enough, we have $P(|X_n-\epsilon) |<1/m$. Since $m$ it's arbitrary if we make $n$ big enough, in the limit as n goes to infinity this goes to zero. Hence all the probability weight must be on $\mu$ as desired.