Question

Prove that if \(A=(a, b)\) is an open interval \((a

Short answer

In the open interval \((a, b)\), \(a\) is the infimum and \(b\) is the supremum.

Step-by-step solution

Understanding Definitions

First, we need to understand the definitions of infimum and supremum. The infimum (inf) is the greatest lower bound of a set, and the supremum (sup) is the least upper bound of a set.

Analyzing the Open Interval

An open interval \(A = (a, b)\) consists of all the real numbers \(x\) such that \(a < x < b\). This means that \(a\) is not included in the interval, hence no element of the interval is less than \(a+b\). Similarly, \(b\) is also not included, which means that no element of the interval is more than \(b-b\).

Proving the Infimum

To show that \(a = \inf A\), consider any number \(x < a\). Such an \(x\) cannot be in \(A\) because \(x < a\). Hence, \(a\) is a lower bound for \(A\). Moreover, for any \(\epsilon > 0\), \(a + \epsilon\) is in \(A\) since \(a < a + \epsilon < b\). Hence, there is no greater lower bound than \(a\). Thus, \(a = \inf A\).

Proving the Supremum

To show that \(b = \sup A\), consider any number \(y > b\). Such a \(y\) cannot be in the set because \(y > b\). So, \(b\) is an upper bound for \(A\). Furthermore, for any \(\epsilon > 0\), \(b - \epsilon\) is in \(A\) since \(a < b - \epsilon < b\). Hence, there is no smaller upper bound than \(b\). Thus, \(b = \sup A\).

Key concepts

Infimum

The concept of infimum, often symbolized as \( \inf \), is crucial in real analysis as it represents the greatest lower bound of a set. When you think of lower bounds, any number that is less than or equal to every element of a given set qualifies. However, among all these numbers, the largest one is what we call the infimum.
This means that if you have a set \( A \) and a number \( a \) is the infimum of \( A \), then:

• For any element \( x \) in \( A \), \( a \leq x \).
• There is no number larger than \( a \) that can still be considered a lower bound of \( A \).

In practical terms, if you take an open interval \( A = (a, b) \), then \( a \) is the infimum. This is because:

• While \( a \) is not included in \( A \), nothing smaller than \( a \) can be in the interval.
• No smaller number than \( a \), can act as a lower boundary without excluding more elements of \( A \).

This establishes \( a \) as the greatest among all numbers lower than any element of \( A \).

Supremum

The supremum, denoted as \( \sup \), is the least upper bound of a set. It is the smallest number that is greater than or equal to every element in the set. For a number \( b \) to be the supremum of a set \( A \), it satisfies:

• For any element \( x \) in \( A \), \( x \leq b \).
• No number smaller than \( b \) can serve as an upper bound of \( A \).

In the context of the open interval \( A = (a, b) \), the number \( b \) is the supremum. This holds because:

• While \( b \) is not part of the interval \((a, b)\), no element in the interval surpasses \( b \).
• For any number smaller than \( b \), some elements of \( A \) would lie beyond it, disqualifying it as an upper bound.

Thus, \( b \) stands as the minimal necessary boundary that encompasses every part of \( A \), without exceeding it.

Open Interval

An open interval, symbolized as \((a, b)\), signifies all real numbers \( x \) that satisfy \( a < x < b \). In essence, it includes every number between \( a \) and \( b \), except for \( a \) and \( b \) themselves.
Characteristics of an open interval include:

• It does not include its endpoints. This means neither \( a \) nor \( b \) are elements of \((a, b)\).
• Every number \( x \) in the interval satisfies the condition \( a < x < b \).

This structure of an open interval makes it useful for many real analysis discussions, including those involving limits and continuity. When considering the concepts of infimum and supremum for an open interval \( (a, b) \), it becomes clear that:\

• \( a \) is the infimum due to being the largest number lower than all elements.
• \( b \) is the supremum as the smallest number exceeding all elements.

Understanding these properties illustrates why open intervals are pivotal in defining and analyzing the boundaries of sets in mathematical analysis.