Cardinality of infinite sequences of $0$ and $1$ $\geq |\mathbb{R}|$

Question

Think of all infinite sequences of $0$s and $1$s. Let the set be $S$. I want to prove that the cardinality $|S|$ is greater than or equal to $|\mathbb{R}|$. I think it is useful to use the fact that the set $T$ of reals in $(0,1)$ has the same cardinality as $\mathbb{R}$. If I can create an injective function from $S$ to $T$ then it would imply $|S|=|T|=|\mathbb{R}|$. I think of taking the the infinite number in $S$, typically 10011001... and map to the number in $T$ with that decimal expansion, so 0.10011001.... Then wouldn't that be an injection?

Answer 1

An injective map from S to $\mathbb{R}$ shows that $\mathbb{R}$ is bigger than S, not vice-versa. But you're in the right ballpark for constructing the map.