Supremum and Infimum

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Supremum and Infimum

• Mika Seppälä

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Distance in the Set of Real Numbers
Definition
Triangle Inequality
Triangle inequality for the absolute value is
almost obvious. We have equality on the right
hand side if x and y are either both positive
or both negative (or one of them is 0).
We have equality on the left hand side if the
signs of x and y are opposite (or if one of
them is 0).
Definition
The distance between two real numbers x and y
is x-y.
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Properties of the Absolute Value
Example
Proof
Problem
When do we have equality in the above estimate?
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Solving Absolute Value Equations
Example
Solution
Conclusion
The equation has two solutions x 2 and x
-3.
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Solving Absolute Value Inequalities
Example
Solution
Conclusion
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Upper and Lower Bounds
Let A be a non-empty set of real numbers.
Definition
A set A need not have neither upper nor lower
bounds.
The set A is bounded from above if A has a
finite upper bound.
The set A is bounded from below if A has a
finite lower bound.
The set A is bounded if it has finite upper and
lower bounds.
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Supremum
Completeness of Real Numbers
The set A has finite upper bounds. An
important completeness property of the set of
real numbers is that the set A has a unique
smallest upper bound.
Definition
The smallest upper bound of the set A is called
the supremum of the set A.
sup(A) the supremum of the set A.
Notation
Example
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Infimum
The set A has finite lower bounds. As in the
case of upper bounds, the set of real numbers is
complete in the sense that the set A has a
unique largest lower bound.
Definition
The largest lower bound of the set A is called
the infimum of the set A.
inf(A) the infimum of the set A.
Notation
Example
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Characterization of the Supremum (1)
Theorem
Proof
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Characterization of the Supremum (2)
Theorem
Proof Contd
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Characterization of the Infimum
Theorem
The proof of this result is a repetition of the
argument the previous proof for the supremum.
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Usage of the Characterizations
Example
Claim
Proof of the Claim
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2
1
2
and