How does the comparison test incorporate the cauchy criterion for convergence?

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- November 12th 2008, 09:14 PMpila0688analysis - 2
**How does the comparison test incorporate the cauchy criterion for convergence?** - December 13th 2008, 03:01 PMMathstud28
Let

So the comparison test says that if for all with and converges, then so does

Proof: converges implies that converges. Now since every convergent sequence is Cauchy it also follows that there exists some that if then . But because of how we defined we see this is equivalent to . But by

And

.

From there we conclude that converges

It is clear that we used the Cauchy convergence criterion at