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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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    You don’t need Schröder-Bernstein: you have $2^\omega\le|[0]|\le 2^\omega$.2012-05-29
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    @Brian: What? That is precisely the situation when you *do* need Schroeder-Bernstein.2012-05-29
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    Not if you’re doing everyday mathematics, which I take to be the context here. The appeal to S-B is incongruous in this setting.2012-05-29
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    Schroeder–Bernstein just is the statement that, if $|A|\le|B|\le|A|$, then $|A|=|B|$. The only situation I can think of when it would be appropriate to use this fact without drawing attention to it is when your audience is sufficiently versed in set theory that they don't need to be told. I do not think that is the case here.2012-05-29
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    I know exactly what it is, and have done for decades. It’s completely irrelevant unless you want to fuss about the axiom of choice; under any other circumstances you can take it as given that cardinalities are well-ordered and hence that if $\kappa\le|A|\le\kappa$, then $|A|=\kappa$.2012-05-29
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    So you prefer to rely on a *harder* result, and use it without explicit mention to people who, quite possibly, know neither its statement nor its proof? That is terrible pedagogy.2012-05-29
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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