Prove that the CDF of a random variable is always right-continuous

Your struggle comes from a lack of understanding of continuity. In order to show $f:\mathbb{R}\to\mathbb{R}$ is (left/right)-continuous, it suffices to show that if $x_n \to x$ monitonically (from the left/right) that $f(x_n) \to f(x)$.

This has nothing to do with probability, it follows from basic real analysis. Try to work out the proof. (It should only be 3-4 lines at most). Start from the definition of continuity.

To be able to properly talk about the notion of continuity for CDFs or PDFs, it is better to use measure theoretic foundation of probability.

But anyway, your argument can be completed as follows. Choose $A_n=\left(-\infty,x+\frac{1}{n}\right]$ as your intervals. As you said, you can prove $F_X\left(x+\frac{1}{n}\right) \to F_X(x)$. Now , for each $\epsilon$, you can find $n$ such that: $$0 \leq F_X\left(x+\frac{1}{n}\right)-F_X(x)<\epsilon.$$ Because the CDF is monotonic, choose $\delta<\frac{1}{n}$. Then for all $y$, with $x<y<x+\delta$ we have: $$0 \leq F_X(y)-F_X(x)\leq F_X(x+\delta)-F_X(x)\leq F_X\left(x+\frac{1}{n}\right)-F_X(x)<\epsilon,$$ which finishes the proof.