# Cantor set and ternary expansions

I'm trying to show that if $x \in [0,1]$ has a ternary expansion consisting only of $0$'s and $2$'s, then $x$ is in the Cantor ternary set. The proofs I've seen typically rely on induction. Is it valid to simply argue that $x$ is removed in the $k$-th iteration of the construction of the Cantor set only if the $k$-th digit in all ternary expansions of $x$ is equal to 1?

For example, if $x \in (\frac{1}{3},\frac{2}{3})$, then all ternary expansions of $x$ must begin with 1.

Yes, if you can show that the set of points removed at $k$th step are those with $1$ in the $k$th digit of (every) ternary expansion, then the conclusion follows.