Explanation of Blaschke's selection theorem

Question

Here's the first part of the proof of Blaschke's selection theorem.

I have two questions that I couldn't figure out:

1. What is the base case $m=1$?

2. For the underlined part, how can we be sure that the ball of radius $1/2^{m+2}$ that contains $x$ also intersects all of the $C_{(m+1)i}$'s? All we know at this point is that the $C_{(m+1)i}$'s are members of the sequence $C_{mi}$'s, and thus are of Hausdorff distance $1/2^m$ from each other. Doesn't this mean the ball containing $x$ needs to have radius at least $1/2^{m+1}$ to intersect all other $C_{(m+1)i}$'s?