Exercise 2.6.5. If (rn) and (yn) are Cauchy sequences, then one easy way to prove that...

 

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Exercise 2.6.5. If (rn) and (yn) are Cauchy sequences, then one easy way to prove that (n + Yn) is Cauchy is to use the Cauchy Criterion. By Theorem 2.6.4, (n) and (yn) must be convergent, and the Algebraic Limit Theorem then implies (n + yn) is convergent and hence Cauchy. (a) Give a direct argument that (n + yn) is a Cauchy sequence that does not use the Cauchy Criterion or the Algebraic Limit Theorem. (b) Do the same for the product (nyn). Exercise 2.6.5. If (rn) and (yn) are Cauchy sequences, then one easy way to prove that (n + Yn) is Cauchy is to use the Cauchy Criterion. By Theorem 2.6.4, (n) and (yn) must be convergent, and the Algebraic Limit Theorem then implies (n + yn) is convergent and hence Cauchy. (a) Give a direct argument that (n + yn) is a Cauchy sequence that does not use the Cauchy Criterion or the Algebraic Limit Theorem. (b) Do the same for the product (nyn).

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