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Let $a$ be an real number and let $S$ be the set of all sequences in $\mathbb{R}$ converging to $a$. What is the Cardinality of $S$?

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1 Answers 1

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First note that there are only $2^{\aleph_0}$ sequences of real numbers. This is true because a sequence is a function from $\mathbb N$ to $\mathbb R$ and we have $\left|\mathbb{R^N}\right|=\left(2^{\aleph_0}\right)^{\aleph_0}=2^{\aleph_0\cdot\aleph_0}=2^{\aleph_0}$

Now note that take any injective sequence which converges to $a$, then it has $2^{\aleph_0}$ subsequences. All are convergent and they all converge to $a$.

Therefore we have at least $2^{\aleph_0}$ sequences converging to $a$, but not more than $2^{\aleph_0}$ sequences over all, so we have exactly $2^{\aleph_0}$ sequences.

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    To make commenter's point even more explicit, for what it's worth, the set $\{(b,a,a,a,\ldots):b\in\mathbb R\}$ already gives the lower bound of $2^{\aleph_0}$.2012-12-30