CRN 10459: HW 3

Problem 11. 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 12. Suppose $(a_n)$ is a bounded sequence such that all of its converging subsequences converge to the same limit, say $L$. Show that $(a_n)$ converges to $L$ as well.

Problem 13. Consider the following two properties:

1. Every non-empty set that is bounded from above has a supremum.
2. Every Cauchy sequence converges.

Show that (2)$\Rightarrow$(1). ((1)$\Rightarrow$(2) was done in class, via the Bolzano-Weierstrass Theorem.)

Problem 14.

1. Show: If $x$ is an accumulation point of $A\cup B$, then $x$ is an accumulation point of $A$, or $x$ is an accumulation point of $B$ (or both).
2. Does the result also hold for a countably infinite collection of sets? Give a proof, or provide a counterexample.

Problem 15. Find all accumulation points of the set \[\left\{\frac{1}{m}+\frac{1}{n}\ |\ m,n\in\mathbb{N}\right\}\] Remember that $A=B\ \Leftrightarrow\ (A\subseteq B)\wedge (B\subseteq A)$.