A question on the proof of the Baire Category theorem

Question

In many proofs (but not all) of the Baire Category theorem one requires on the n:th induction step that: $\overline{B(y_n,r_n)} \subset U_n \cap B(y_{n-1},r_{n-1})$ (where $\{ U_n \}_{n \geq}$ is a sequence of dense open sets). I wonder if this is a necessary step? Isn't it be enough to require: $B(y_n,r_n) \subset U_n \cap B(y_{n-1},r_{n-1})$ , and then show that the limit of $\{y_n\}$ belongs to $B(y_n,r_n) $?

Answer 1

The reason (probably, I don't know what text you're using) is because to get the non-empty intersection, Cantor's theorem can be applied: if $C_n$ is a sequence of decreasing closed sets such that $\operatorname{diam}(C_n) \to 0$ then in a complete metric space $\cap_n C_n \neq \emptyset$. The $\overline{B(y_n, r_n)}$ can be used as $C_n$ provided the $r_n$ decrease to $0$, as $\operatorname{diam}(C_n) \le 2r_n$ in this case.