Cdf of a discrete random variable and convergence of distributions

[Artusartos]

In the page that I attached, it says "...while at the continuity points x of $F_x$ (i.e., $x \not= 0$), $lim F_{X_n}(x) = F_X(x)$." But we know that the graph of $F_X(x)$ is a straight line y=0, with only x=0 at y=1, right? But then all the points to the right of zero should not be equal to the limit of $F_{X_n}(x)$, right? Because $F_X(x)$ is always zero at those points, but $F_X(x)$ is 1? So how do I make sense of that?

Thanks in advance

 

[Stephen Tashi]

But we know that the graph of $F_X(x)$ is a straight line y=0, with only x=0 at y=1, right?

No, I think $F_X(x)$ is the cumulative distribution, not a density function.

 

[Artusartos]

No, I think $F_X(x)$ is the cumulative distribution, not a density function.

Oh, ok...

But it's still confusing. What if n=4 (for example)? Then $$F_{X_n} = 1$$ if $$x \geq 1/4$$, and $$F_{X_n}=0$$, when $$x < 1/4$$, right? So for any x between 0 and 1/4, the limit at those points is 0, but the limit of $$F_X$$ at those points is 1...so the limits are not equal, are they?

 

[Stephen Tashi]

So for any x between 0 and 1/4, the limit at those points is 0,

What limit are you talking about? Something like $lim_{x \rightarrow 1/8} F_{X_4}(x)$ ? I see nothing in the discussion in the book that dealt with that sort of limit. The limits under consideration involve $n \rightarrow \infty$.