I'm reading the following lemma in L. Tu book on manifolds. There is a statement in this proof that I didn't see that in this book before. That is, for any open set $U$ in $M$ and a point $p∈U$, there is a coordinate open set $U_α$ such that
$$p∈U_α⊂U.$$
I don't know how the author has concluded this statement. Could anyone explain this for me, please? Here is the lemma and it's proof in book.
Lemma. A manifold $M$ has a countable basis consisting of coordinate open sets.
Proof. Let ${(U_α,φ_α)}$ be the maximal atlas on $M$ and $\mathcal B=\{B_i\}$ a countable basis for $M$. For each coordinate open set $U_α$ and point $p∈U_α$, choose a basic open set $B_{p,α}∈\mathcal B$ such that
$$p∈B_{p,α}⊂U_α.$$
The collection $\{B_{p,α}\}$, without duplicate elements, is a subcollection of $B$ and is therefore countable.
For any open set $U$ in $M$ and a point $p∈U$, there is a coordinate open set $U_α$ such that
$$p∈U_α⊂U.$$
Hence,
$$p∈B_{p,α}⊂U,$$
which shows that $\{B_{p,α}\}$ is a basis for $M$.
Take a chart $(U_\beta, \varphi_\beta)$ around $p$. Then $(U_\alpha, \varphi_\alpha) \doteq (U_\beta \cap U, \varphi_\beta\big|_{U_\alpha \cap U})$ is again a chart around $p$, whose domain satisfies $U_\alpha\subseteq U$.