Determine if this set is compact or not

[Johnson04]

Homework Statement

Let E be the set of all x\in [0,1] whose decimal expansion contains only the digits 4 and 7. Is E compact? Is E perfect?

Homework Equations

The Attempt at a Solution

My answer is: E is compact and perfect.

By Heine-Borel theorem (E is compact equivalent to E is closed and bounded), since E is bounded, and \forall x\in E, and \epsilon > 0, we always have that the neighborhood of x, N_\epsilon (x) contains infinitely many elements of E, in other words, any number in E is a limit point of E. Conversely, I assume E is not closed, that is, there exists at least one number which is not in E, but it is a limit point of E. I denote this number by y. Suppose the i-th digit of y, say y_i is the first digit which is not in \{4,7\}. Then we always can find some other numbers which are between y and the closest number in E. So we can conclude that if y\not\in E, y must not be a limit point of E. So we can conclude that E is closed and perfect. By the aforementioned Heine-Borel theorem, it is also compact.

However, I am still worrying about this proof, since I am not very sure on the correctness of the proof. Can you guys help me to see if both my answer and my proof are correct? Thanks a lot!

 

[Dick]

Your answer is correct. It's a Cantor type set. Your proof leaves a lot to be desired. And it is a bit complicated since you have to prove two things, that E is closed and perfect. Start with closed. As you said, if a point x is not in E, then it has a digit yi that is not 4 or 7. You have to show x is a positive distance away from E. Can you give an estimate for a positive lower bound on how far away it might be? Suppose the digit is say '3'?