# Replacing continuous variables in a limit with a sequence

I have a question regarding the nuts and bolts involved in the proof of the limit of CDFs. The statement is that

Proposition: Let $$X$$ be a random variable with CDF $$F_X(.)$$. Then $$F_X(.)$$ posses the following property.

$$\lim_{x \to \infty} F_X(x) = 1$$

Proof:

Consider a sequence $$\{x_n\}$$ with $$n \in \mathbb{N}$$ such that it monotonically increases to $$\infty$$. Then we have

$$\begin{eqnarray} \lim_{x \to \infty}F_X(x) &=& \lim_{x \to \infty} \mathbb{P}(X \leq x) \\ &=& \lim_{n \to \infty} \mathbb{P}(X \leq x_n) \label{eqnref} \\ &=& \mathbb{P} \left \{ \bigcup_{n \in \mathbb{N}} \{ω : X(ω) ≤ x_n \} \right \} \\ &=& \mathbb{P}(\Omega) \\ &=& 1. \end{eqnarray}$$

My question is regarding the second step where the continuous variable $$x$$ is replaced by the member of a sequence $$x_n$$. I feel lack of rigor in this step. To be precise, my questions are

1. Why is this step valid?
2. The trajectory that $$x$$ can take while approaching $$\infty$$ are many, while the sequence $$x_n$$ is assumed to be monotonically increasing. How do we know for sure that this difference in the way to approach infinity will not change the limit?
3. Is there a way to make the proof look more rigorous as in is there a rigorous way to substantiate this step of replacing $$x$$ with $$x_n$$?

2. I understand your point of multiple trajectories for $$x$$, but take into account that for any of these multiple trajectories, you can extract a monotonically increasing one.
$$\lim_{x \to \infty} \mathbb{P}(X \leq x) = l \iff \forall \epsilon >0, \exists A \mid x > A \Rightarrow | \mathbb{P}(X \leq x) - l | < \epsilon$$
$$\lim_{n \to \infty} \mathbb{P}(X \leq x_n) = l \iff \forall \epsilon >0, \exists N \mid n > N \Rightarrow |\mathbb{P}(X \leq x_n) - l | < \epsilon$$
So, what you need is to find a way from a $$A$$ (resp $$N$$) large enough to a have the nice property, to find a $$N$$ (resp $$A$$) large enough to have the other nice property. In order to do so, I would write the definition of : $$\lim_{n \to \infty} x_n = +\infty$$.