Exercise 2, § 2

A countable dense subset A of R is not closed. [If (a_n) is a sequence obtained by arranging the points of A in some order, define a sequence of intervals [b_n, c_n] such that b_{n-1} < b_n < c_n < c_{n-1} for all n, and such that [b_n, c_n] contains no point a_k with index k ≤ n; then use Theorem 2.] Deduce that R is uncountable (cf. § 8, no. 6).