Another counting problem

The set $S$ has the same size as the interval $[0,1]$ , which is uncountable. We get an intuition for why they are the same size when you represent each $x \in [0,1]$ in binary and consider the mapping $f:S \rightarrow [0,1]$ by $\{a_1, a_2, a_3, \ldots \} \mapsto 0.a_1a_2a_3\ldots _2$ (the subscript denoting binary digit notation), which is almost a bijection. I say "almost" because some distinct sequences like $\{0,1,1,1,\ldots \}$ and $\{1,0,0,0,\ldots \}$ map to the same number in binary; namely $0.1_2$ which has the alternative representation $0.0111\ldots _2$ . Nonetheless, the spirit of this argument can be made formal since there are only countably many of these points.