Define $F(a-)=\lim _{x\uparrow a}F(x)$. Then, if $F$ is nondecreasing, $F(a-)=\lim _{n\rightarrow\infty} F(a-1/n)$. Use (1.1) to show that if a random variable $X$ has cumulative distribution function $F_X$, that $P(X<a)=F_X(a-)$. Also show that $P(X=a)=F_X(a)-F_X(a-)$.
$B_1\subset B_2 \subset \dots \Rightarrow \mu\left(\bigcup_{n=1}^\infty B_n\right)=\lim_{n\rightarrow\infty} \mu(B_n)$ (1.1)
Here is a relevant definition.
If $(\mathcal {E, B}, P)$ is a probability space, where $e\in \mathcal E$ are called outcomes, $B\in\mathcal B$ are called events, and $P(B)$ is called the probability of $B$,
$$\begin{split}F_X(a-)&=\lim_{n\rightarrow\infty} F_X(a-1/n)\\ &=\lim_{n\rightarrow\infty} P(\{e\in\mathcal E:X(e)\le a-1/n\})&&\text{ definition of cdf}\\ &=P\left(\bigcup_{n=1}^\infty \{e\in\mathcal E:X(e)\le a-1/n\}\right)&&\text{ from (1.1)}\\ &=P\left(\{e\in\mathcal E:X(e)<a\}\right)\\ &=P(X<a)\end{split}$$
And for the next part,
$$\begin{split}F_X(a)-F_X(a-)&=P(X\le a)-P(X<a)\\ &=P(\{e\in \mathcal E: X(e)\le a\})-P(\{e\in \mathcal E: X(e)< a\})\\ &=P(\{e\in\mathcal E: X(e)=a\})&&\text{ def. of measure}\\ &=P(X=a)\end{split}$$
Does it seem correct? In particular, for the second part, I can't directly go from step 1 to step 4 even though it might be obvious from what a probability is, I think, since all I know is that $P$ is a probability measure. And I also was wondering if using $P(\{e\in\mathcal E:X(e)\dots\})$ is necessary.