What is the cardinality of the class of $0$ in $\mathbb{R}$? In other words: what is the cardinality of the class of all rational Cauchy sequences that converge to $0$?
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HINT: If $q_n\to 0$, then given any infinite subset of $\Bbb N$, $A$, replacing $q_n$ for $n\in A$, with $0$ is again a Cauchy sequence converging to $0$.
Asaf Karagila
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i don't quite see where you're going with this... – Mathematicxcz Dec 17 '14 at 10:08
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@Mathematicxcz If you take for example $q_n=\frac1n$ (or any sequence of rational numbers convergent to 0 such that $q_n\ne0$), then Asaf's hint should help you show that there is at least $\mathfrak c=2^{\aleph_0}$ such sequences. (There is at least as many sequences in your equivalence class as the number of subsets of $\mathbb N$.) On the other hand, it is a standard exercise in cardinal arithmetic to show that the set of all rational sequences has cardinality $\mathfrak c$. – Martin Sleziak Dec 17 '14 at 11:54
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And maybe it is worth mentioning that the same argument works for real sequences (instead of rational sequences). See also here. – Martin Sleziak Dec 17 '14 at 12:01