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Re: Please check my proof
Posted:
Oct 10, 1999 2:28 PM


C.D. <*@*> wrote in message <1T2M3.2760$UG5.195380@typ11.nn.bcandid.com>... >For homework we were asked to prove the following: > >Suppose that K is a nonempty subset of R that is not compact. Prove that >there is a monotone sequence in K that does not converge to a point in K. > >Now, this is what I've got.... > >K not compact implies that there exists a sequence {x_n} in K such that >there exists a subsequence {x_n_k} of {x_n} such that {x_n_k} does not >converge to any point x in K (By definition of compact).
That's not quite right, though it's true. It should be: There exists a sequence {x_n} in K such that for any subsequence {x_n_k} of {x_n}, {x_n_k} does not converge to a point in K. It's important to learn how to negate statements. "for all" becomes "there exists" and "there exists" becomes "for all".
Since {x_n_k} is a >subsequence, by definition it is monotone and it is contained in {x_n} which >implies it is contained in K.
I don't see how it's monotone.
Therfore {x_n_k} is a monotone sequence in K >that does not converge to a point in K. > >I quess what I'm worried about is that a subsequence does not count as a >sequence here.
A subsequence is definitely a sequence. Now if you could find a monotone subsequence of {x_n}, then you could use my generaliztion of your statement above to finish the proof, since you know that no subsequence of {x_n} converges to a point in K.
Chuck Cadman



