Is this a valid proof of Lemma (2.1.6) (c) form Grimmett & Stirzaker that "distribution function F is right-continuous"?
Plug definitions $F(X) = \mathbb{P}(x \leq X)$ to what right continuity means
$F(h + X) \rightarrow F(X) $ as $h \downarrow 0$:
$\lim_{h \to 0} \mathbb{P}(x \leq h + X) $ should be equal to $\mathbb{P}(x \leq X)$
Set $\{x \leq h + X\}$ is disjoint union of
$\{x \leq X\} \bigcup \{X < x \leq h + X\}$
so $$ \lim_{h \to 0} \mathbb{P}(\{ x \leq X\} \bigcup \{X < x \leq h + X\}) = \lim_{h \to 0}\mathbb{P}(\{x \leq X\}) + \lim_{h \to 0} \mathbb{P} (\{X < x \leq h + X\})$$
because $\lim_{} $ and $\mathbb{P}$ are additive, then first limit is constant $\mathbb{P}(\{ x \leq X\})$ since it doesn't depend on h and second is
$$\mathbb{P}_{h \to 0}(\{X < x \leq h + X\}) \rightarrow \mathbb{P}(x = X) = 0$$
since its probability of single point so $\mathbb{P}(\{ x \leq X\}) + 0 = \mathbb{P}(x \leq X)$ as required.
P.S. Grimmett & Stirzaker suggest to use Lemma (1.3.5) which says that when $A_1,..$ is increasing sequence of events $A_1 \subseteq A_2 \subseteq …$ and A is their limit:
$$A = \bigcup A_i = \lim_{i \to \infty} A_i$$
then $\mathbb{P}(A) = \lim_{i \to \infty} \mathbb{P}(A_i)$
I only noticed that after I came up with proof in question.
Best Answer
The equality $$\mathbb{P}_{h \to 0}(\{X < x \leq h + X\}) \rightarrow \mathbb{P}(x = X) = 0$$ follows from Lemma(1.3.5), so your proof is almost correct. It remains to deduce limit from that Lemma.