Let $\{(U_\alpha,\phi_\alpha)\}$ be the maximal atlas on a Manifold $M$. For any open set $U$ in $M$ and a point $p \in U$, prove the existence of a coordinate open set $U_\alpha$ such that $p \in U_\alpha \subset U$.
By definition of an atlas $M = \bigcup_{\alpha = 1}^n U_\alpha$.
Attempt at proof: Let $U \subset M$ be any open set. Suppose $U = \bigcap_{\alpha =1}^{n-1} U_\alpha$. Now let $p \in U$ be an arbitrary point. Then $p \in \bigcap_{\alpha =1}^{n-1} U_\alpha$.
This is where my confusion starts since I'm not sure how to obtain the $U_\alpha \subset U$ containment. I'm also puzzled by whether I should define $U$ as a union of a certain amount of coordinate open sets. I reason that since $U$ is an open set in $M$ then $U$ contains some amount of coordinate open sets which then enables the existence of that coordinate open set.
I would appreciate hints and advice to help me navigate this problem.