Baire Category Theorem: Question About Countable Dense Open Sets

[mr.tea]

Hi,
I have a (probably stupid) question about the Baire Category Theorem. I am looking at the statement that says that in a complete metric space, the intersection of countable many dense open sets is dense in the metric space.
Assume that we have the countable collection of dense open sets $\{U_n\}$ in a complete metric space $X$, and let $x \in X, \epsilon>0$. Since $U_1$ is dense in $X$, there is $y_1\in U_1$ with $d(x,y_1)<\epsilon$. Also, as $U_1$ is open, there is $r_1>0$ with $B(y_1;r_1)\subset U_1$. Then, we can arrange $r_1<1$ such that $\overline{B(y_1;r_1)} \subset U_1\cap B(x;\epsilon)$.
Now my question is why we can arrange that the closure will be contained in each of them? I think intuitively it sounds correct, but I didn't succeed to prove it rigorously. Can you please help me here?

 

[mathwonk]

there is a typo in your statement, the intersection is dense, not open.

 

[mr.tea]

there is a typo in your statement, the intersection is dense, not open.

Sorry, fixed it.

 

[mathwonk]

i guess you mean by "each of them" the two open sets in your discussion. this is trivial. if a point lies in an open set in a metric space then it has some positive distance from the outside of that set, and hence the ball of radius 1/2 that distance has closure entirely contained in that set.