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Recall that the real number $0$ is defined as the class of all rational Cauchy sequences that converge to $0$. How can I determine the cardinality of this class?

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There are only $\aleph_0^{\aleph_0}$=$2^{\aleph_0}$ rational sequences in total, so clearly there are at most that many rational sequences converging to $0$.

For a lower bound, if $A$ is any subset of $\mathbb{N}$, let $x^A$ be the sequence $x^A_n=\begin{cases}1/n & n \in A \\ 0 & n \not \in A \end{cases}$ Different $A$s give different sequences, so this gives us $2^{\aleph_0}$ rational sequences converging to $0$.

Thus by Schroeder–Bernstein there are exactly $2^{\aleph_0}$ such sequences.

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    Yes, because it’s part of the normal background of everyday mathematics. I prefer not to engage in set-theoretic pedantry when it’s not really appropriate $-$ and I say that as a set-theoretic topologist with an inclination to indulge in it. The point is that in terms of everyday mathematics **no result is needed**. Sure, if you want to dig into the set theory there’s much to be said, but it’s not something that most people using basic cardinal arithmetic need or need to worry about. Dragging it in unnecessarily is bad pædagogy.2012-05-29