Set Theory Question

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confusedfool
Posts: 1
Joined: Tue Feb 11, 2014 1:53 am UTC

Set Theory Question

Postby confusedfool » Tue Feb 11, 2014 3:56 am UTC

Assuming "D" is a set of sequences of natural numbers with the property that for any sequence of natural numbers "s" there exists a sequence "d" in the set "D" and a number "n0" from the set of natural numbers such that d(n)>s(n) for any n>n0.

Prove that any such set D is uncountable.

Can anyone help me? I have no idea how to prove this.

lalop
Posts: 210
Joined: Mon May 23, 2011 5:29 pm UTC

Re: Set Theory Question

Postby lalop » Thu Feb 13, 2014 4:18 am UTC

Assume, for contradiction, that D is countable, so that you can list the elements of D as something like this:

1 -> 1, 2, 3, 4, 5, ...
2 -> 200, 10, 5, 8, 9, ...
3 -> 88, 90, 2, 4, 6, ...
...

Working "diagonally", find a sequence whose tail eventually gets bigger than all of these.

Spoiler:
Ensure that the nth number of your constructed sequence is greater than the nth number of sequences 1 to n.


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