## Limits of Sums, Differences, Products and Quotients of Series

Sums, products, differences and quotients of sequences converge in the obvious way.

Theorem

If converges to and converges to then converges to Proof: Choose There is a positive integer such that if then and a positive integer such that if then Let then if  and then Hence converges to Theorem

If converges to and converges to then converges to Proof: Choose Since there is a positive integer such that if then a positive integer such that if then and a positive integer such that if then Let then if   and  so take then  Theorem

If converges to and converges to then converges to Proof: Choose There is a positive integer such that if then and a positive integer such that if then Let then if  and then Hence converges to Theorem

If converges to and converges to then converges to Proof: Choose Since converges, it is bounded so there exists such that for all Let There is a positive integer such that if then and a positive integer such that if then Let then if  and then Hence converges to  