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

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?

• Have you ever heard of numbers written in binary before? Can you find a bijection between the set of real numbers in $[0,1]$ written in binary and your set $S$? – JMoravitz Nov 30 '16 at 19:12
• That is a indeed nice approach, I didn't think about binary numbers, so used to our decimal system... – user30523 Nov 30 '16 at 19:37
• Are we sure the decimal expansion is unique, for example 0.999...=1.0000...., but since we are in $(0,1)$ it is fine? – user30523 Nov 30 '16 at 19:49
• That is a good observation to make. No, the decimal expansion is not technically unique, but you could instead show an injection from $[0,1)$ to $S$ by taking the binary expansion of the numbers where if given a choice between two representations, take the one with repeated zeroes instead of repeated ones. This would at least show $|\Bbb R|\leq S$. To show the other direction, then perhaps you can be a bit trickier, making those sequences which end in infinitely many repeated zeroes to the binary number in $[0,1)$ and with infinitely many repeated ones to numbers in $[1,2]$ – JMoravitz Nov 30 '16 at 19:57
• @JMoravitz: If you want to inject $S$ into $\mathbb{R}$ just consider things like 0.11111110111... as decimal expansions rather than binary ones. – Daniel McLaury Nov 30 '16 at 20:55

## 1 Answer

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.

• Of course, it is the other way around. It does not matter for bijections but for injections the direction is crucial. – user30523 Nov 30 '16 at 19:35