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I don't know how "many" (cardinality) sub-sequences are there in a sequence.

Or equivalently,

What is the cardinality of the set of countably infinite subsets of a countably infinite set?

I guess it should not be $\aleph_0$, maybe $2^{\aleph_0}$ ($\aleph_0$ is the cardinality of natural numbers).

Thank you.

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    it's $2^{\aleph_0}$, by definition.2012-06-02
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    Use the fact that if $A$ is infinite and $B$ countable, the cardinality of $A\cup B$ is the same as the cardinality of $A$. You will indeed find $2^{\aleph_0}$.2012-06-02
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    @Dunstan: Not by definition, since the OP is asking about $|[\omega]^\omega|$, not $|\wp(\omega)|$.2012-06-02
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    How many subsequences does $a_n=1$ has?2012-06-02
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    @BrianM.Scott point taken.2012-06-02

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