Exercise 2.6.3. If (xn) and (yn) are Cauchy sequences, then one easy way to prove that (xn + yn) is Cauchy is to use solution th
e Cauchy Criterion. By Theorem 2.6.4, (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 (xn n).
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