HW-8_solutions

HW-8_solutions - Homework 8 Solutions December 1 2008 12.2...

This preview shows pages 1–2. Sign up to view the full content.

Homework 8. Solutions December 1, 2008 12.2 Show: lim sup | s n | = 0 if and only if lim s n =0 Proof Suppose lim sup | s n | =0. Let ± > 0, so there is N such that k > N implies sup {| s n | : n > k } < ± . This implies that n > N we have | s n | < ± , which is just the deﬁnition of lim s n = 0. Now suppose lim | s n | = 0. Let ± > 0, there exist N so that n > N implies | s n | < ± . So for all k > N sup {| s n | : n > k } < ± , which implies that lim sup | s n | = 0. 12.4 Show that: lim sup s n + t n lim sup s n + lim sup t n for bounded sequences s n and t n . Give an example of two sequences for which the equality does not hold. Proof Let ( s n ) and ( t n ) be bounded sequences. Then, the sets { s n : n N } , { t n : n N } and { s j + t k : j,k N } are bounded. Moreover, for each N N the subsets { s n : n > N } , { t n : n > N } and { s n + t n : n > N } are bounded (and are nonempty), by Completeness Axiom the subsets have a supremum. Let s N = sup { s n : n > N

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

This document was uploaded on 10/01/2009.

Page1 / 3

HW-8_solutions - Homework 8 Solutions December 1 2008 12.2...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online