Taking limit of a sequence inside discrete distribution function

Question

Suppose that there exists a positive, integer sequence $\{c_k\}$, satisfying $c_k\geq c^*$, such that $c_k\to c^*$. Further suppose that $X$ is a binomial random variable with size $n$ and parameter $p$. Now, $$ P(X>c_k)=P(X>c^*)-P(c^*<X\leq c_k) $$ I'm interested in the limit $$ \lim_{c_k\downarrow c^*}P(c^*<X\leq c_k) $$ Does this limit exist?

Context

I'm trying to show that $P(X>c_k)\to P(X>c^*)$, but since $X$, and hence $P(X>x)$ is discrete, I don't think that the limit can pass inside so nicely. Rather I think the difference between $P(X>c_k)$ and $P(X>c^*)$ should involve some local limit (i.e. a gap jump) like $P(X=c^*)$. However, I'm not sure if I'm overthinking it or can show that this local limit is indeed present.

Answer 1

Note that $c_k$ is a sequence of positive integers, hence for there exists $N$ such that $k > N \implies c_k=c^*$.

Hence for $k>N$, $Pr(c^* < X \le c_k)=Pr(c^* < X \le c^*)=0$.

We have $$\lim \limits_{c_k \downarrow c^*}Pr(c^* < X \le c_k)=0$$

Answer 2

For any random variable $X$ we have $P\{X=c^{*}\}=\lim P\{c_k<X\leq c^{*}\}$ if $c_k \leq c^{*}$ and $c_k \to c^{*}$. (This follows from the fact that $\{c^{*}\}=\cap_k (c_k,c^{*}]$ and continuity property of the measure induced by $X$).