Exercise 2.6.3

If $\left(x_n\right)$ and $\left(y_n\right)$ are Cauchy sequences, then one easy way to prove that $\left(x_n+y_n\right)$ is Cauchy is to use the Cauchy Criterion. By Theorem 2.6.4, $\left(x_n\right)$ and $\left(y_n\right)$ must be convergent, and the Algebraic Limit Theorem then implies $\left(x_n+y_n\right)$ is convergent and hence Cauchy.

(a)

Give a direct argument that $\left(x_n+y_n\right)$ is a Cauchy sequence that does not use the Cauchy Criterion or the Algebraic Limit Theorem.

(b)

Do the same for the product $\left(x_n{y}_n\right)$.