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## Homework Statement

If a linear ordered set P has a countable dense subset, then

[itex] |P| \leq 2^{\aleph_0} [/itex]

## The Attempt at a Solution

because it has a linear order then all elements of P can be compared x<y .

And because it is dense that mean that I can find an element f

such that x<f<y for any x or y.

so we can partition this set into countable many pieces.

now either their are countably many elements between

any two elements. And the union of countably many things

with a countable number of objects is countable.

Or their are an uncountable number of things in between

any of the 2 dense elements, but this union would

be [itex] 2^{\aleph_0} [/itex] if anything larger was in between

then the cardinality would not work.

This is a little informal, just want to know if this is on the right track.