As part of a HW assignment in the course elementary set theory, I was given the following question:
Prove that the set of all binary sequences (sequences of $0$ and $1$) except for the binary sequences with two consecutive zeros has cardinality $\aleph$.
Hint: define an injective function from a set with cardinality $2^{\aleph_0}$ to the given set. then use CSB theorem.
Let us call the given set A and the set of all binary sequences B.
$A\subseteq B$ thus $\vert A\vert \leq \vert B\vert$ and i know that $B=\{0,1\}^\mathbb N$ and that $\vert\{0,1\}^\mathbb N\vert=\aleph$
Therefore I have that $\vert A\vert \leq \aleph$.
For the second inequality, I thought of using the hint by finding an injetive function from the set $B$ to the set $A$ but I can't find one.
Can anyone help me find an injective function from B to A?
I will also appreciate a different solution that uses the hint above.