Well, I thought about it more and here is my solution:
We will construct a subset as follows: At each kth step, restrict the subset to values which do not have a digit 4 appearing in the kth digit slot of their respective decimal expansions. Thus, after k steps, what remains is segments of length . On the other hand, at each individual jth step, we remove segments of length , and therefore by the kth step we have removed a total of segments.
We condense these facts into the two following observations in the limit as :
(Total length remaining).
(Total length removed).
We conclude therefore that .
Indeed, it is evident this is true for any similar set since where d = # of digits in chosen expansion (binary, ternary, decimal, etc.).