Larry Wasserman , in his All of Statistics, states that for any cdf $F$ :
$F$ is non-decreasing
$\lim_{x\rightarrow -\infty} F = 0 \ \ \text{and} \ \ \lim_{x\rightarrow \infty} F = 1$
$F$ is right continuous
He then writes that proving that if $F$ satisfies these conditions, then it is a cdf for some random variable, requires "some deep tools in analysis."
Does anyone know where I can find a proof of this exact statement, and if not, how it would be proved?
Thanks.