Exercise 9, § 8

Prove that the triadic set of Cantor $K$ (IV, p. 9) is identical to the set of real numbers $x \in (0, 1)$, whose development to base 3 (resp. improper development to base 3 if $x$ is origin of an interval contiguous to $K$) $x = \sum_{n=1}^\infty u_n 3^{-n}$ is such that $u_n \neq 1$ for every $n$ (hence $u_n = 0$ or $u_n = 2$). If one then sets $v_n = u_n/2$ for every $n$, and $f(x) = \sum_{n=1}^\infty v_n 2^{-n}$, prove that $f$ is a continuous surjective mapping from $K$ onto $(0, 1)$; deduce that $K$ has the power of the continuum.