If (xn) and (yn) are Cauchy sequences, then one easy way to prove that (xn + yn) is Cauchy is to use the Cauchy Criterion. By Theorem, (xn) and (yn) must be convergent, and the Algebraic Limit Theorem then implies (xn + yn) is convergent and hence Cauchy.
(a) Give a direct argument that (xn + yn) is a Cauchy sequence that does not use the Cauchy Criterion or the Algebraic Limit Theorem.
(b) Do the same for the product (xnyn).