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If you know that a given set O is countable.

$\#O\leq \#\mathbb{N}$

Does this imply that the following statement holds?

$\# O \leq \#\mathbb{R}$

I'm not sure, but I think it makes sense, because $\mathbb{N}\subset\mathbb{R}$ and therefore you can construct easily an injective function between $\mathbb{R}$ and $\mathbb{N}$.

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Yes, it does, and your reasoning is correct: you have an injection $f:O\to\Bbb N$, and the map $g:\Bbb N\to\Bbb R:n\mapsto n$ is an injection, so the composition $g\circ f:O\to\Bbb R$ is also an injection.

In fact it implies that $\# O < \#\mathbb{R}$, since $\Bbb R$ is uncountable.

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    Sorry---I should have said $\le$.2012-09-07