CRN 11378: HW 3

Problem 11. Let $X$ be a non-empty set that is bounded from below. Show that there is a decreasing sequence $(x_n)_{n=1}^\infty$ of elements in $X$ that converges to $\inf X$.

Problem 12. 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 13. Let $(a_n)_{n=1}^\infty$ be a Cauchy sequence, and let $\varphi:\mathbf{N}\to\mathbf{N}$ be a one-to-one function. Show that the sequence $(a_{\varphi(n)})_{n=1}^\infty$ is a Cauchy sequence.

Problem 14. 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 15. 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-Weierstraaa Theorem.)