Understanding Limit Points and Compact Sets in Real Analysis
School
Southern New Hampshire University**We aren't endorsed by this school
Course
MAT 470
Subject
Mathematics
Date
Dec 11, 2024
Pages
3
Uploaded by Dejahnc1998
MAT 470 β Module 3 AssignmentSouthern New Hampshire UniversityDejah Crews-SilerNovember 13, 2024
3.2.2Let ? = {(β1)π+2π: ? = 1, 2, 3, β¦ } ??? ? = {? β ?: 0 < ? < 1}= {β1 + 2, 1 +22, β1 +23, 1 +24, β1 +25, 1 +26,. . . . . .}= {1,2, β0.33,1.5, β0.60,1.33}a)What are the limit points?The limit points are -1 and 1 b)Is the set open? Closed?Set A is not a closed set. Set A does not have interior points, and therefore is not an open set. Set A is not open or closed set. c)Does the set contain any isolated points?1 is an isolated point in the set of A d)Find the closure of the set. ?Μ
= ? βͺ ?(?) = {(β1)π+2?: ? = 1, 2, 3, β¦ } βͺ {β1}3.3.2a) NN is a set of all natural numbers and is not compact b) ? β© [0,1]Having a set of rational numbers that are intersected with [0,1] = ? β© [0,1]this making ? β©[0,1]not compact c) The cantor set A cantor set has an intersection that is of a closed set, because there is a subset of [0,1] it makes it bounded making cantor set compact. d){1 + 122+132+. . . +1π2: ? β π}Because limπββ(1 +122+132+. . . +1π2) = β(1π2)βπ=1it is convergent and therefore the sum isnβt a set, and it is not compact.
e) {1,12,23,34,45, . . . }Because limπββ(ππ+1) = 1this makes the only limit point a given set in which this limit is between 0 and 1 that makes it bounded. Therefore, this set is compact.3.4.2Does there exist a perfect set consisting of only rational numbers? - Having a set of all rational numbers that are between [0,1]. We can take ? = ? β© [0,1], this set contains all limit points. Irrational numbers in an interval [0, 1] are the limit point of the set ? β©[0,1]. In this set every point is considered a lime point because between having 2 rational number. 3.5.5Show that it is impossible to write where for each ? β π, ?πis a closed set containing no nonempty open intervals. We assume ?πβ?? ?? ????? π????π??. πΏ?? ππ= β?πππ=1?πis closed and ?πππ? ???? 3.5.9Decide whether the following sets are dense in R, nowhere dense in R, or somewhere in between.a)? = ? β© [0. 5]?π??? ?= [0,5] ?βπ?β π? ? ???????? ???? π???????, ?βπ? ????? ?β?? π? ??? ??? ?? ????? b)? = {1π: ? β π}Closure of given set ? = {1π: ? β π} βͺ 0, which does not contain any non open interval therefore it is not dense c)The set of irrationals Let the interval ?, ? β ? ?β??? ? < ? ?β?? ?β??? ??π?? π????π???? ?????? ? βπ
π???β ?β?? ? < ? < ?d)The Cantor set This set is nowhere dense, that if C contains an open interval (a, b) then |?| β₯ |?, ?| = |?, ?|?? ?? ???? ?β?? |?| = 0. ?β??????? ?βπ? ??? ?????? ?????π? ??? ???? π???????.