Terence Tao uncountability of $\mathbb{R}$ There is a small detail I would like to understand. In the proof presented by Tao below: I don't understand why do we have the following cancellation:
$\Sigma_{n < n_0 : n \in A} 10^{-n} - \Sigma_{n < n_0 : n \in B} 10^{-n}$ ? 
I mean we could have elements $n \in \mathbb{N}$ such that $n < n_0$ for which $n \in A - B$ or vice versa. 

Our definition of $n_0$ is such that it is the least $k$ such that $k$ is in set $A$ but not $B$ or vice versa. Thus, all $n$ below $n_0$ would appear in both sets, so we can just cancel them.
