Of the following theorem (Zorich, Mathematical Analysis II, p. 18):
I'm trying to understand the proof of the $ \Leftarrow $ implication. Before showing the proof in question, I post here a simple lemma that will be used in it:
The proof of the implication mentioned above is shown below, where what confuses me is highlighted in red.
What guarantees me that even in $ \widetilde {B} (a_1,1) $ it is true that for any sequence with points inside it, one can extract always a convergent subsequence that has its limit inside $ \widetilde {B} (a_1,1) $?