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I am trying to show the completeness of R, using the LUB property.

Problem is that I don't know, given a Cauchy sequence,where the limit would come from; I can check if a sequence {an} converges to a specific value, but I don't know how to come up with the limit value that the sequence would converge to.

I imagine as the intervals containing the terms am (m>N) become smaller, as |am-ak|N, maybe an converges to the limiting point of intervals (am,ak).

I used limsup and iminf to show that an converges , but this does not tell me what an should converge to.

Any Suggestions?

1 Answers 1

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Sure.

Step 1: Show that any Cauchy sequence is bounded.

(This is true in any metric space and has nothing to do with completeness.) It follows that the limsup and liminf of (the underlying set of terms of) your sequence are finite.

Step 2: Show that any sequence with limsup $L < \infty$ has a subsequence converging to $L$.

Step 3: Now you have something specific to try to show the Cauchy sequence converges to: namely, $L$. Show in fact that if a subsequence of a Cauchy sequence converges to some $L$, then the sequence itself converges to $L$.

Step 4 (optional): Think about what would happen if in Steps 2 and 3 you chose to use the liminf instead of the limsup.

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    @Andres: okay, but no one is going to give you any points for that! :)2011-02-04