Prove that the sum of two Cauchy sequences is Cauchy without using:

Let {x_n} be a sequence of real numbers. Then {x_n} ic Cauchy iff {x_n} converges.

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- Sep 25th 2007, 05:17 PMtaypezCauchy
Prove that the sum of two Cauchy sequences is Cauchy without using:

Let {x_n} be a sequence of real numbers. Then {x_n} ic Cauchy iff {x_n} converges. - Sep 25th 2007, 05:29 PMJhevon
- Sep 25th 2007, 05:45 PMtaypez
That would lead me to use the Cauchy Theorem which it said not to and since the Cauchy Thm uses Bolzano-Weierstrass, I'm assuming I can't use that either.

- Sep 25th 2007, 05:54 PMJhevon
- Sep 25th 2007, 06:00 PMThePerfectHacker
- Sep 25th 2007, 08:25 PMJhevon
Bored.

Let and be Cauchy sequences.

Define . We show that is Cauchy.

Since is Cauchy, for all , there exists an such that implies .

Similarly, since is Cauchy, we can find an such that implies

Now, fix such an , and choose . Then implies that:

Thus, is a Cauchy sequence

QED.