Detemine cardinality of $ \{f \mid f: \mathbb{N} \rightarrow [0,1] \}$.
We can show ran$(f) \neq [0,1] $ by a diagonal argument. But I am not sure how to use this to determine the cardinality.
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I think you haven't read the problem carefully. No particular function $f$ can be surjective. but you're not asked about particular functions. You want the cardinality of the set of all the functions. Hint: each particular function can be thought of as a sequence of real numbers in the unit interval. – Ethan Bolker Dec 19 '18 at 17:12
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1Some answers that address your question are given here – Ben Grossmann Dec 19 '18 at 17:16