probability - Necessary and Sufficient Conditions for a CDF

This is an attempt to prove Theorem 1.5.3. in Casella and Berger. Note that the only things that have been proven are really basic set-theory with $\mathbb{P}$ (a probability measure) theorems (e.g., addition rule). Recall for a random variable $X$, we define $$F_X(x) = \mathbb{P}(X \leq x)\text{.}$$

Theorem. $F$ is a CDF iff:

1. $\lim\limits_{x \to -\infty}F(x) = 0$


2. $\lim\limits_{x \to +\infty}F(x) = 1$


3. $F$ is nondecreasing.


4. For all $x_0 \in \mathbb{R}$, $\lim\limits_{x \to x_0^{+}} F(x)= F(x_0)$

$\Longrightarrow$ If $F$ is a CDF of $X$, by definition,
$$F_{X}(x) = \mathbb{P}(X \leq x) = \mathbb{P}\left(\{s_j \in S: X(s_j) \leq x\} \right) $$

where $S$ denotes the overall sample space.

$(3)$ is easy to show. Suppose $x_1 \leq x_2$. Then notice
$$\{s_j \in S: X(s_j) \leq x_1\} \subset \{s_j \in S: X(s_j) \leq x_2\}$$
and therefore by a Theorem,
$$\mathbb{P}\left(\{s_j \in S: X(s_j) \leq x_1\}\right) \leq \mathbb{P}\left(\{s_j \in S: X(s_j) \leq x_2\}\right)$$
giving $F_{X}(x_1) \leq F_{X}(x_2)$, hence $F$ is nondecreasing.

I suppose $(1)$ and $(2)$ aren't consequences of anything more than saying that $\{s_j \in S: X(s_j) \leq -\infty\} = \varnothing$ and $\{s_j \in S: X(s_j) \leq +\infty\} = S$ (unless I'm completely wrong here). But this seems to suggest that $$\lim_{x \to -\infty}\mathbb{P}(\text{blah}(x)) = \mathbb{P}(\lim_{x \to -\infty}\text{blah}(x))$$
where $\text{blah}(x)$ is a set dependent on $x$. At this point of the text, this hasn't been proven (if it's even true).

I'm not sure how to show $(4)$.

$\Longleftarrow$ I don't know how to prove sufficiency. Casella and Berger state that this is "much harder" than necessity, and we have to establish that there is a sample space, a probability measure, and a random variable defined on the sample space such that $F$ is the CDF of this random variable, but this isn't enough detail for me to go on.