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An elusive sequence...
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Author Topic: An elusive sequence...  (Read 9091 times)
John
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« on: February 21, 2010, 10:45:26 AM »

Who can think of a sequence X_n such that for every real number L there is a subsequence of X_n that converges to L?

So that sequence must have subsequences converging to every real number.... I stayed up all night trying to think of possible solutions, but this is out of my league I think now! Huh
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« Reply #1 on: February 21, 2010, 12:08:03 PM »

Perhaps it is not that bad. A sequence just means it must be a countable set so you are looking for a countable set that is 'dense', or in simpler terms kind of 'everywhere'. That might be a well known set, and since it is countable you can think of it as a sequence. If you find that set/sequence the rest might might not be too bad. Good luck!
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« Reply #2 on: February 21, 2010, 01:55:53 PM »

How can a set be countable and dense, it would not have a one to one relation to the set of Natural Numbers?

I really really appreciate your help, friend.  Smiley
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« Reply #3 on: February 21, 2010, 07:05:26 PM »

Check http://www.mathcs.org/analysis/reals/infinity/countble.html for sets and sequences that are countable yet contain intuitively "a lot" of numbers.
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