Show that the sets D and E of Exercise 5 have the same cardinality.
Solution
Recall \[ D=(\mathbb Z_+)^\omega, \qquad E=\{0,1\}^\omega. \] There is an injection \(E\to D\) given by \[ (x_n)\mapsto(x_n+1). \]
For the reverse direction, encode \(a=(a_1,a_2,\ldots)\in D\) by the binary sequence \[ \underbrace{11\cdots1}_{a_1}0\, \underbrace{11\cdots1}_{a_2}0\, \underbrace{11\cdots1}_{a_3}0\cdots. \] Because every \(a_i\ge1\), this sequence contains infinitely many separator zeros, and the lengths of the successive blocks of ones uniquely recover \(a_1,a_2,\ldots\). Thus this encoding is injective \(D\to E\).
By Schroeder–Bernstein, \[ \boxed{|D|=|E|}. \]