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Replies: 5   Last Post: May 30, 2003 2:40 AM

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 Chuck Cadman Posts: 16 Registered: 12/12/04
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.

Date Subject Author
10/10/99 C.D.