Baby Rudin Theorem 3.7 Clarification

The sequence is choosen inductively. First he picks $n_1$ so that $p_{n_1}\neq q$ and set $\delta = d(p_{n_1}, q)$. Then he consider $\delta/2$. As $q$ is a limit point, there is $n_2$ (which can be chosen so that $n_2 > n_1$) so that

$$d(p_{n_2}, q) < \delta /2$$

Inductively, he choose $n_k>n_{k-1}> \cdots n_2 >n_1$ so that

$$d(p_{n_i}, q)< \frac{\delta}{2^i}.$$