4130hw6

# 4130hw6 - 4130 HOMEWORK 6 Due Thursday April 1(1 Let A R A...

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4130 HOMEWORK 6 Due Thursday April 1 (1) Let A R . A point x A is called isolated if it is not a cluster point of A . (a) Can an open set have an isolated point? Can a closed set have one? (b) Give an example of a countable set with no isolated points. (2) Section 3.3.1 # 8. (3) Section 4.2.4 # 3. (Recall that an interval is, by deﬁnition, a subset I of R such that for all x,y I and all z R with x < z < y , we have z I .) (4) In this question, we will show that every positive real number has an n th root. (a) Let x (0 , ) and n N . Show that there exist α,β R with α n < x < β n . (b) Show that there exists y R with x = y n . (c) For x [0 , ), show that there exists a unique y [0 , ) with x = y 2 . We denote this y by x . (d) Deﬁne f : [0 , ) R by f ( x ) = x . Show that f is a continuous function. (5) Two monasteries, A and B , are joined by exactly one path AB which is 20 miles long. One morning, Brother Albert (a monk) sets out from monastery

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4130hw6 - 4130 HOMEWORK 6 Due Thursday April 1(1 Let A R A...

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