Cardinal equality: $\;\left|\{0,1\}^{\Bbb N}\right|=\left|\{0,1,2,3\}^{\Bbb N}\right|$

Question

I need to prove the above equality without Cantor-Bernstein Theorem or cardinals arithmetic (i.e., a bijection must be found).

I know that for example $\;S\to 1_S=\;$ the indicator function, gives a bijection $\;P(\Bbb N)\to \{0,1\}^{\Bbb N}\;$ , so if I can find a bijection $\;P(\Bbb N)\to\{01,2,3\}^{\Bbb N}\;$ then I can compose these two and that's all. Yet this last one is making problems to me, so any help will be appreciated.

Answer 1

HINT: You can interpret a binary sequence as a sequence base four by grouping it into pairs of digits.