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