Hello, I am having some difficulty on problem 17 of chapter 2 in Rudin's Principles of Mathematical Analysis. The problem reads: "Let E be the set of all x in [0,1] whose decimal expansion contains only the digits 4 and 7. Is E compact?"
My problem is that there is a theorem in this book which says that if every infinite subset of a set E has a limit point in E, then E is compact.
I used the following to justify that E is compact:
By the way, I denote the neighborhood of a point, p, of radius r as N(p, r).
Let F be an infinite subset of E and let s be a finite set of 4's and 7's with m digits. Since F is infinite, F contains every combination of n 4's and 7's, so s is an element of F.
Then there exist an s1 and s2 such that |s1 - s2| < 3 X 10^-m.
s1 and s2 are therefore limit points since s1 is in N(s2, x) and s2 is in N(s1, x) for any x > 3 X 10^-m. This x can always be found since m is a natural number with no upper bound. Therefore, every infinite subset of E contains a limit point in E, and E is compact.
However, the following seems to be an open cover of E for which there is no finite subcover:
F(n) = (0, .7) where F(1) = (0, .7), F(2) = (0, .77) F(3) = (0, .777), and so on. This would seem to imply that E is not compact .
I'm sure I'm making some silly mistake, but I would really appreciate it if someone could help me sort out what I'm doing wrong.