Exercise 9, § 8

Show that Cantor's triadic set $K$ (§ 2, no. 5) is the set of all real numbers $x \in [0, 1]$ whose triadic expansion (resp. improper triadic expansion if $x$ is the origin of an interval contiguous to $K$)

is such that $u_n \neq 1$ for each $n$ (so that $u_n = 0$ or $u_n = 2$). If $v_n = \frac{1}{2} u_n$ for each $n$, and if $f(x) = \sum_{n=1}^{\infty} v_n 2^{-n}$, show that $f$ is a surjective continuous mapping of $K$ onto $[0, 1]$; deduce that $K$ has the power of the continuum.