# Let $F_X(x):=P(X\leq x)$ a distribution function of a random variable $X$. Prove that $F_X$ is right-continuous.

Let $F_X(x):=P(X\leq x)$ a distribution function of a random variable $X$.

Prove that $F_X$ is right-continuous.

I need to show that for every non-increasing sequence $x_n$ with $\lim x_n=x$ I will get:

$$\lim_{n\to\infty}f(x_n)=f(x_0)$$

How do I show this ? Any ideas ?

-
I believe in your statement $x_0=x$ – gt6989b Mar 8 '13 at 19:55
This is indeed equivalent to the (reverse) monotone convergence of finite measures. – Sangchul Lee Mar 8 '13 at 20:04

Pick a set $S_n =\{X \leq x_n\}$. If $x_n$ is decreasing so is $S_n$ and $\{X \leq x_n\} = \cap_1^\infty S_n$. Now take the limit for $F(x_n)$ and you should be set.

-

Hint: Define $C_{n}$ as the event that $$x<X\leq x+\frac{1}{n}$$ and note $$P(C_{n})=F_{X}(x+\frac{1}{n})-F_{X}(x)$$

And that $$C_{1}\supseteq C_{2}\supseteq C_{3}...$$

What is $$C=\cap_{n\in\mathbb{N}}C_{n}$$ ?

What is $$\lim_{n\to\infty}P(C_{n})$$ ?

Also recall $F$ is monotone to complete the proof.

-