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 2.6.4, (xₙ) and (yₙ) must be convergent, and the Algebraic Limit Theorem then implies (xₙ + yₙ) is convergent and hence Cauchy. Give a direct argument that (xₙ + yₙ) is a Cauchy sequence that does not use the Cauchy Criterion or the Algebraic Limit Theorem.