CRN 11982: HW 4

In Problems 16 and 17 do not use the fact that Cauchy sequences are convergent sequences.

Problem 16. Suppose $(a_n)$ is a Cauchy sequence, and that $(b_n)$ is a sequence satisfying $\lim_{n\to\infty} |a_n-b_n|=0$. Show that $(b_n)$ is a Cauchy sequence.

Problem 17. Let $(a_n)_{n=1}^\infty$ be a Cauchy sequence, and let $\varphi:\mathbb{N}\to\mathbb{N}$ be a one-to-one function. Show that the sequence $(a_{\varphi(n)})_{n=1}^\infty$ is a Cauchy sequence.

Problem 18. A Cauchy sequence $(a_n)$ is said to be positive, if for all $k\in\mathbb{N}$ there is an $N\in\mathbb{N}$ such that $a_n>-\frac{1}{k}$ for all $n\geq N$.

1. Show that the sum of two positive Cauchy sequences is positive.
2. Show that the product of two positive Cauchy sequences is positive.

Problem 19. Give a proof of the Alternating Series Theorem (Theorem 2.7.7).

Problem 20. Exercise 2.7.6