Countability of sets, usage of cantor set

Question

How could I prove that the set $S_1$ of all infinite sequences that consists of 0's and 2's, implies $S_1$ is uncountable. Could I say $S_1$ is equivalent to the cantor set?

Answer 1

Yes, you could. You could also show that the map $$\phi:S_1\to 2^{\Bbb N}\\ \phi(f)= \frac12f$$ is a bijection.