A function discontinuous on a countably infinite set This is a proof from Royden's Real Analysis and there is a part that I do not understand( Underlined in red). Any explanation is much appreciated. 



To put it in simply, consider a point $x_0$ in the set $(a,b) \sim C$. Now take a neighborhood about $x_0$ and notice that if you take a neighborhood small enough you can ensure that no points $q_1 ,q_2, \ldots ,q_{n}$ are in this neighborhood. Therefore if $x$ is a point in the said neighborhood then $|f(x_0)-f(x)|$ will be at most the tail end of the series $\sum \frac{1}{2^n}$ from the point $\frac{1}{2^{n+1}}$ onward. And this tail end sum can be shown to be less than $\frac{1}{2^n}$
