4.1 Prove that in any metric space (S, d) every closed ball S,[xo] is a closed set. 4.2 Let F be a closed set for k = 1, 2,..., n in (S, d). Show that Fr is closed. k=1
4.1 Prove that in any metric space (S, d) every closed ball S,[xo] is a closed set. 4.2 Let F be a closed set for k = 1, 2,..., n in (S, d). Show that Fr is closed. k=1
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.1: Postulates For The Integers (optional)
Problem 26E: Prove that the cancellation law for multiplication holds in Z. That is, if xy=xz and x0, then y=z.
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2.1 Let B be the set of all bounded sequences of real numbers and define the function d :B × B ! R by
d(x, y) = supn|xn − yn|.
Show that (B, d) is a metric space.
4.1 Prove that in any metric space (S, d) every closed ball Sr[x0] is a closed set. (5)
4.2 Let Fk be a closed set for k = 1, 2, . . . , n in (S, d). Show that
n
Uk=1
Fk is closed.
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