The cardinality of all the infinite binary sequences that don't contain 010

Question

Find the cardinality of all the infinite binary sequences that don't contain 010

I think it's $\aleph_0$. I marked the set all infinite binary sequences that don't contain 010 in A, and the set of All the infinite binary sequences that don't contain 01 in B. $B\subseteq A$, and $|B|=\aleph_0$, therefore $\aleph_o\le|A|$. But I don't know how to show that $|A|\le\aleph_o$ . Thanks

Answer 1

This is not $\aleph_0$, but rather $2^{\aleph_0}$, as it contains all 0-1 sequences of the form \begin{equation} x_1x_1x_2x_2x_3x_3x_4x_4\ldots \end{equation} where $x_i\in\{0,1\}$.

Answer 2

There is a bijection from the set of all binary sequences to the set of all sequences not containing 010: just replace all occurrences of 01 with 011. And the cardinality of the set of all binary sequences is...?