= Almost-sure limit of independent random variables is constant
If <independent random variables> $X_n$ converge almost surely to a finite $X$, then $\limsup_nX_n$ is measurable for the <tail sigma-algebra>. The <Kolmogorov zero-one law> forces every rational threshold event for that limit to have probability zero or one. A real-valued <probability distribution> with this property is a point mass, so $X$ is almost surely constant. Consequently the <Second Borel-Cantelli lemma> upgrades almost sure convergence of independent sequences to <complete convergence>.
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