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The answer by this post started by stating that every random variable has a cumulative distribution function (CDF).

Question: How to prove that the existence of CDF of every random variable?

I understand that some random variable might not have probability density function.

But I have no idea how to prove the question.

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It is not a matter of proof but merely of definition.

If $X$ is a random variable then we can define the function $\mathbb R\to\mathbb R$ as prescribed by: $$x\mapsto P(X\leq x)$$

Observe that - because $X$ is a random variable - for every $x\in\mathbb R$ the set $\{X\leq x\}$ is a measurable set so that the expression $P(X\leq x)$ makes sense and is an element of $[0,1]\subseteq\mathbb R$.

This function is labeled as the CDF of $X$ and is mostly denoted as $F_X$.

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