econstruct the proof of the following statement: The supremum of a bounded from above subset A of R is unique. Proof. We do a proof by . Assume that x₁ and x2 are both supremum of A with x₁ for all e> 0 there exists x EA such that x x₁.This proves that x₁ is not an x2. Since x₂ = . It follows that there exists x EA such that x our hypothesis. upper bound contradiction sup(A) # X_2-x_1>0 equal to x2. We can assume that x₁ X2-6. Choose, e for A in different from with
econstruct the proof of the following statement: The supremum of a bounded from above subset A of R is unique. Proof. We do a proof by . Assume that x₁ and x2 are both supremum of A with x₁ for all e> 0 there exists x EA such that x x₁.This proves that x₁ is not an x2. Since x₂ = . It follows that there exists x EA such that x our hypothesis. upper bound contradiction sup(A) # X_2-x_1>0 equal to x2. We can assume that x₁ X2-6. Choose, e for A in different from with
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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Transcribed Image Text:Reconstruct the proof of the following statement:
The supremum of a bounded from above subset A of R is unique.
Proof.
We do a proof by
Assume that x₁ and x2 are both supremum of A with x₁
for all e> 0 there exists x E A such that x
x₁.This proves that x₁ is not an
x2. Since x₂ =
It follows that there exists x EA such that x
our hypothesis.
upper bound
contradiction sup(A)
#
X_2-x_1>0
equal to
X2. We can assume that x₁
X2 - E. Choose, e
for A in
different from
with
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