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Recall that a countable set $S$ implies that there exists a bijection $\mathbb{N}\to S$. Now, I consider (0,1). I want to prove by contradiction that $(0,1)$ is not countable.

First, I assume the contrary that there exists a bijection $f$, and I can find an element in $S$, but not in the range of $f$. But I can't find such element. How can you construct such $f$?

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    The title of your question and the body ask two different things. Do you want to show the set of all functions from $\mathbb{N}$ into $\{0,1\}$ is uncountable or the open interval $(0,1)$ is uncountable?2012-04-10

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