Decreasing Sequence of Events in Statistics There is a question from lecture notes whose answer I do not quite understand. The question is 

Let $X$ have a cdf $F$. Let $x$ be any point and suppose $x_n$ is a decreasing sequence such that $x_n \rightarrow x$ as $n \rightarrow \infty$. Show $F(x_n) \rightarrow F(x)$ as $n \rightarrow \infty$.

The solution according to lecture notes is:
If $x_n \downarrow -\infty$ Then $\lim_{m\to\infty}F(x_m) = \lim_{m\to\infty}P(X \leq x_m) = P(\lim_{m\to\infty}(X \leq x_m)) = P(\bigcap_{m}(X \leq x_m)) = P(\varnothing) = 0$ Hence  $\lim_{x\to -\infty}F(x) = 0$ which can be written as $F(-\infty) = 0$.
Then this implies the conclusion.
Can someone help me understand how this argument answers the question or suggest an alternative answer? 
 A: The way you've phrased it, it could work at most for $x=-\infty$.
Probability (at least the way it is conventionally axiomatized) is countably additive.  That means if $A_1,A_2,A_3,\ldots$ are a pairwise disjoint sequence of events, then $P(A_1\cup A_2\cup A_3\cup\cdots)$ $=P(A_1)+P(A_2)+P(A_3)+\cdots$.
Finite additivity tells us that
$$
\begin{align}
F(a)+P(a<X\le b)
& =P(X\le a)+P(a<X\le b) \\[8pt]
& = P(X\le a\text{ or }a<X\le b)\quad\text{(finite additivity)} \\[8pt]
& = P(X\le b) \\[8pt]
& = F(b),
\end{align}
$$
whence
$$
P(a<X\le b)=F(b)-F(a).
$$
Now suppose $x_n\downarrow x$.  Then
$$
\begin{align}
F(x) & = P(X\le x) = 1 - P(X>x) \\[8pt]
& = 1-P\Big( X>x_1\text{ or }x_1\ge X>x_2\text{ or }x_2>X\ge x_3\text{ or }\cdots \Big) \\[8pt]
& = 1 - P(X>x_1) - \sum_{i=1}^\infty P(x_i>X\ge x_{i+1})\quad\text{(countable additivity)} \\[8pt]
& = F(x_1) - \sum_{i=1}^\infty (F(x_i)-F(x_{i+1})) \\[8pt]
& = F(x_1) - \lim_{n\to\infty} \sum_{i=1}^n (F(x_i)-F(x_{i+1})) \\[8pt]
& = \lim_{n\to\infty} F(x_1)-F(x_1)+F(x_2)-F(x_2)+\cdots+F(x_{n-1})-F(x_{n-1})+F(x_n) \\[8pt]
& = \lim_{n\to\infty}F(x_n).
\end{align}
$$
A: Cumulative distribution functions are necessarily continuous since they are the integral of a probability density function. So, we have that $F$ is a continuous function which satisfies the theorem from general topology that if $x_{n} \rightarrow x$ as $n \rightarrow \infty$ and $F$ is continuous, then $F(x_{n}) \rightarrow F(x)$ as $n \rightarrow \infty$. This is a much more natural and intuitive solution than the one presented in your lecture notes.
