Let $X$ be the set of all numbers in $[0,1]$ which admit a binary representation $0.c_1c_2c_3...$ such that for every natural number $n$: $c_n=0$ or $c_{n+1}=0$. Prove $X$ is compact and it has measure zero.
I know that this looks like the Cantor set but I cannot express $X$ as intersection of nested closed sets.