Document Type

Thesis

Date of Award

5-1996

School/College

College of Science, Engineering, and Technology (COSET)

Degree Name

MS in Mathematics

First Advisor

Robert M. Nehs

Abstract

A set is dense in R+ if and only if given every open interval in R+ contains elements of M. In this paper encountered with the problems of finding out countable dense subsets in R+. In particular, if am and bn are monotone and diverge to 00 then under what conditions is M = {am/bn I m,n E N} dense in R+. Let {am} and {bn} be two positive increasing sequences, both diverging to infinity. If such that lim m+= am+1/am = I, then M = {am/bn I m,n E N} is dense in R+. The following specific results were obtained when lim m+= am+1/am = r > 1 and lim (1) M = {rm/sn I m,n E N} is dense in R+ if and only 1 2 if In(s)/ln(r) lS irrational where r,s > 1. (2) M = {mrm/nsn I m,n E N} is dense in R+ if and only'Rf x = In(s)/ln(r) is irrational where r,s > 1. (3) If r,s > 1 such that In(s)/ln(r) is rational, then M = {mrm/sn I m,n E N} is dense in R+. (4) M = {m!/2n I m,n E N} is dense in R+. Results in this paper were used to prove that {sin(n) In = 1,2,3 ...} and {cos(n) In = 1,2,3 ...} are each dense in [-l,lJ

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