or each föllowing stätements, ermine whether it is true or false. Give a counterexample IF IT IS FALSE (but you do not need to prove it if it is true). (a) A bounded above set of real numbers always has a maximum. (b) A nonempty bounded above set of real numbers always has a supremum. (c) A nonempty bounded above set of integers always has a supremum and a maximum.
or each föllowing stätements, ermine whether it is true or false. Give a counterexample IF IT IS FALSE (but you do not need to prove it if it is true). (a) A bounded above set of real numbers always has a maximum. (b) A nonempty bounded above set of real numbers always has a supremum. (c) A nonempty bounded above set of integers always has a supremum and a maximum.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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For each of the following statements, determine whether it is true or false. Give a counterexample IF
IT IS FALSE (but you do not need to prove it if it is true).
(a) A bounded above set of real numbers always has a maximum.
(b) A nonempty bounded above set of real numbers always has a supremum.
(c) A nonempty bounded above set of integers always has a supremum and a maximum.
(d) If neither the set A nor the set B is dense in R, then AU B is not dense in R.
(e) A set that is dense in R must have an infinite number of members.
(f) A set that is dense in R does NOT have a supremum.
(g) Let S = {0.001 · n | n € Z}. Then S is dense in R.
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