I'm trying to solve the exercise in Section 2.4 of Singer & Thorpe, which is to prove that if $S$ is a compact Hausdorff topological space and $(U_n)_{n \in \Bbb N}$ be a family of dense open sets, then $\bigcap_n U_n \neq \emptyset$. Actually, a similar result is shown (Theorem 2.4.2) where $S$ is a complete metric space. In its proof one constructs a Cauchy sequence $(s_n)_n$ such that $s_n \in \bigcap_{k=0}^n U_n$, whose limit turns out to reside in $\bigcap_n U_n$.
In solving the exercise, I naturally imitate the proof of Theorem 2.4.2. I can actually construct $(s_n)_n$ such that $s_n \in \bigcap_{k=0}^n U_n$, but I don't know how to make it convergent, because I don't know the "sizes" of open sets that $s_n$ resides in. I would be grateful if you suggest a hint to the exercise.