So I'm trying to show that these two sets have the same cardinality, i.e. there is a possible bijection between the two. I'm trying to use the Cantor-Schroeder-Bernstein theorem as I can't explicitly think of a bijection that will work. For this I need to find an injective map each way. I can find an injective mapping between $R$ and the set of infinite sequences of natural numbers. E.g., for each real number $x$, I can associate it with the sequence $S$, where the first element is $2^{n}$ (where $n$ is the greatest integer less than or equal to $x$) if x is nonnegative, and $3^{n}$ if x is negative. Then the rest of the sequence can be the decimal expansion of x in single digits. E.g. the sequence associated with pi through the function would be $(2^{3}, 1,4, etc)$. This is injective, so the first half of Cantor-Schroeder-Bernstein is satisfied. If I can find an injective function going the other way then I am done.
I first thought of something involving turning the digits of the sequence into a real number, but that failed to be injective as my function would not be able to differentiate between the sequences $(1,0,1,0,1,0,…)$ and $(10,10,10,…)$.
Any hints or help? I'm trying to figure it out without resorting to proofs based on cardinal arithmetic or related to the power set of N.
Best Answer
You need a separator to cure the problem you identified. One way is to write all the numbers in the sequence in binary and put a $2$ between them, so $4,3,5,1,9$ would become $10021121012121001$. Put a decimal point in front and you are done.