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November 8th, 2007, 09:38 AM   #1
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Cardinality of the set of sequences of non negative integers

Let S be the set of all sequences of non-negative integers. What's the cardinality of S? I was told it equals the cardinality of the set of all functions from R to R that sends rationals into rationals, but couldn't prove this

If, for each p >=2, we define S_p as the set of all sequences of non-negative integers bounded by p, then each (a_1, a_2....a_n...) of S_p can be seen as the p-adic representation of the number 0.a1 a2...a_n..., so that to each element of S_p there corresponds one elemet of [0,1]. This proves the cardinalty of S_p, and therefore of S, is at least Aleph(1), but this conclusion is not very interesting.

Does anyone have a hint?
Thank youy
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November 8th, 2007, 10:52 AM   #2
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Re: ardinality of the set of sequences of non negative integ

Quote:
Originally Posted by Artur
Let S be the set of all sequences of non-negative integers. What's the cardinality of S?

[...]

Does anyone have a hint?
Consider sequences like 3,1,4,1,5,9,2,6,5,3,5,8,9,7,9,3,2,3,8,.... There are clearly at least as many sequences as real numbers. Then consider adding one to each member and treating it as a continued fraction.
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