Question

Assignment
1. Investigate the existence of the two iterated limits and the double
of the double sequence f defined by
a) S(o) -
b) = (-1)​

Answer 1

Answer:

Last updatedAug 15, 2020

4.10.E: Problems on Arcs, Curves, and Connected Sets

4.11.E: Problems on Double Limits and Product Spaces

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Contributed by Elias Zakon

Mathematics at University of Windsor

Publisher: The Trilla Group (support by Saylor Foundation)

Table of contents

Given two metric spaces (X,ρ1) and (Y,ρ2), we may consider the Cartesian product X×Y, suitably metrized. Two metrics for X×Y are suggested in Problem 10 in Chapter 3, §11. We shall adopt the first of them as follows.

Definition

By the product of two metric spaces (X,ρ1) and (Y,ρ2) is meant the space (X×Y,ρ), where the metric ρ is defined by

ρ((x,y),(x′,y′))=max{ρ1(x,x′),ρ2(y,y′)}

for x,x′∈X and y,y′∈Y.

Thus the distance between (x,y) and (x′,y′) is the larger of the two distances

ρ1(x,x′) in X and ρ2(y,y′) in Y.