= Helly selection for distribution functions
{c}
A tight sequence of <probability measures> on the real line has a subsequence whose <cumulative distribution functions> converge at every continuity point of a <cumulative distribution function>. Choose a diagonal subsequence converging at every rational $q$, write those limits as $f(q)$, and define $F(x)=\inf_{q>x}f(q)$. Monotonicity gives right continuity, and <tightness of probability measures> gives limits zero and one at the two ends of the real line. Bracketing by rational continuity approximations proves convergence at the continuity points of $F$. The corresponding <probability measure> is the weak subsequential limit: approximate a bounded continuous test function by step functions on a compact interval, and use tightness to bound the tails.
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