$F$ is the cumulative distribution function, having the property of right continuity

The question here is exactly, proving that for the increasing function F it suffices to restrain the study on the sequence $x\_n = x\_0 + \frac{1}{n}$ to prove the right continuity of $F$ , so this is not a question for the proof , this question followed the proof which used $x\_n = x\_0 + \frac{1}{n}$

copied quick hints , on one way we can do it (may contain errors) :

> 
> for a given h>0 , we note $A\_n=]a, a+h/n]$
> use the theorem of the continuity
> applied on the decreasing set
> 
> 
> retraining to $x\_n = x\_0 + \frac{1}{n}$ is not sufficient in the general case we need to
> prove it for any increasing sequence (a counter example is the limit of the
> indicator function of rationals at $0$)
> 
> 
> We have $An= \cup\_{ j= n}^{\infty} A\_j \setminus A\_{j+1} $ and so:
> $P(A\_n)=\sum\_{j=n}^{+\infty}P(A\_j \setminus A\_{j+1} ) $
> 
> 
> $P(A\_j \setminus A\_{j+1} )$ is then the general term of a converging
> series: $P(A\_1)=\sum\_{j=1}^{\infty}P(A\_j \setminus A\_{j+1})$
> 
> 
> the remainder of this serie is $P(A\_n)$ which tends to zero , but
> $P(A\_n)=F(a+h/n)-F(a)$
> 
> 
> 

I don't really understand this, or the pertinence of the use of the indicator function of rationals, I know in general it doesn't suffice to restrain to the sequence $x\_n = x\_0 + \frac{1}{n}$ and we need to prove it for any increasing sequence, so my question is how to prove it is sufficient to do that in this case?