# How to intuitively understand convergence in distribution?

I have this question:

Let $$\{X_n\}$$ be positive integer valued random variables. Prove that

$$X_n\xrightarrow{d}X_0$$ iff for every $$k\geq0$$, $$P[X_n=k]\rightarrow P[X_0=k]$$.

Suppose $$X_n$$ has distribution $$F_n$$. Suppose $$X_n\xrightarrow{d}X_0$$. Given $$k$$ and $$\epsilon < 1/2$$, we have $$(k-\epsilon, k + \epsilon)$$ is an interval of continuity of $$F_0(x)$$ and so
$$P[X_n=k]=P[X_n\in(k-\epsilon, k + \epsilon)]\rightarrow P[X_0\in(k-\epsilon, k + \epsilon)]=P[X_0=k]$$
Why $$(k-\epsilon, k + \epsilon)$$ is an interval of continuity of $$F_0(x)$$? If $$X_n$$ be integer valued, $$F_n$$ should have many jumps. Right?
• Interval of continuity should simply mean that $\mathbb{P}(X_0=k-\varepsilon)=\mathbb{P}(X_0=k+\varepsilon)=0$, which is true, because the jump point of $F_0$ lies in between, at $x=k$. In general for some probability measure $\mu$ on a metric space we call $A$ a set of continuity for $\mu$ if $\mu(\partial A)=0$. This property means that $\mu_n\to \mu$ weakly implies that $\mu_n(A)\to \mu(A)$. This is the reason for the choice of terminology. – WoolierThanThou Dec 5 '19 at 16:20