$[0,1]$ without numbers which have the digit $8$ in them?

Question

I have an interesting task and I simply have no idea how to prove this.

Suppose we have the $[0,1]$ interval, but we remove all numbers, which contain the digit $8$. Prove, that the remaining set is a null set(Lebesgue measure=0).

Any ideas? Help is appreciated :)

Answer 1

Try that (it's a bit rudimentary but it works):

• cut your interval in 10
• isolate the segment $[0.8,0.9[$, whose length is $0.1$ (and is obviously in the complementary of your set)
• for each of the other 9 sub-segments, the sub-sets with no 8 are similar, so you can focus on for instance $[0, 0.1[$
• but now your problem is just homothetic to the initial one
• you should get an equation satisfied by your measure $m$