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The definition of a Cumulative Distribution Function $(CDF)$ says that $$P(X \le x) = F(x)$$

This is all good.

Then my text book gives the following theorem without proof: $$P(X \lt x) = F(x-)$$

The book says that the proof is easy, but with my rusted calculus skills, I have trouble to even intuitively understand this theorem.

I tried to sketch a proof, to improve my understanding: $$F(x-) = \lim_{n \to \infty} F(x - \frac{1}{n}) = \lim_{n \to \infty} P(X < x - \frac{1}{n}) = P(X < x).$$

I assume we can't say $P(X \le x - \frac{1}{n})$, because then $X(\omega) \ge x$, for some $\omega \in \Omega$.

Is this the correct reasoning?

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    If you have $F(x) = P(X\leq x)$ in the definition, you cannot change it to $F(x) = P(X < x)$. So you should have "$\leq$" instead of "<" in $P(X < x-\frac{1}{n})$.2010-11-01

3 Answers 3