I have been reading the book Modern Real Analysis by Ziemer and have come to an exercise in the chapter on measure theory that I am having trouble with. The exercise concerns a Cantor-like set where instead of the middle $\frac{1}{3}$ being deleted on each iteration the middle $\frac{\alpha}{3}$ is deleted with $\alpha \in (0,1)$. Let $C$ be the Cantor set and $C_{\alpha}$ be the Cantor-like set.
The exercise asks the reader to prove that $C_{\alpha}$ is nowhere dense, has cardinality $c = 2^{\aleph_{0}}$, and Lebesgue measure $1 - \alpha$. I am able to show that $C_{\alpha}$ is nowhere dense and that its Lebesgure measure is $1 - \alpha$, but I am having trouble finding a way to prove that $C_{\alpha} = c$. Can someone help with this ?