Problem 1 - Analysis 1: Ein Lehr- und Arbeitsbuch für Studienanfänger

Question

Wann ist eine reelle Zahl \(K^{*}\) keine obere Schranke einer Teilmenge \(A \subseteq \mathbb{R}\) ?

Step-by-Step Solution

Answer

\(K^*\) is not an upper bound if there exists an \(x \in A\) such that \(x > K^*\).

Step 1: Understanding Upper Bounds

To determine when a real number \(K^*\) is not an upper bound for a subset \(A \subseteq \mathbb{R}\), we need to understand that a real number \(K^*\) is considered an upper bound for \(A\) if every element \(x\) in \(A\) satisfies \(x \leq K^*\). To find when \(K^*\) is not an upper bound, the opposite must be true.

Step 2: Identify the Condition

For \(K^*\) to not be an upper bound, there must exist at least one element \(x\) in the subset \(A\) such that \(x > K^*\). This is the key condition: \( \exists x \in A \text{ such that } x > K^* \).

Step 3: Illustrative Example

Consider \( A = \{1, 2, 3\} \) and test \( K^* = 2 \). While 1 and 2 satisfy the condition \( x \leq 2 \), the number 3 does not, since \( 3 > 2 \). Therefore, \( K^* = 2 \) is not an upper bound for \(A\).

Step 4: Conclusion

Thus, a real number \(K^*\) is not an upper bound for a subset \(A\) if, and only if, there exists at least one element in \(A\) that is greater than \(K^*\).

Key Concepts

Real Numbers

Real numbers are an integral part of mathematics and they encompass both rational and irrational numbers, providing a complete version of the number system we use in everyday life.
Real numbers include:

• Whole numbers like 0, 1, 2,...
• Fractions or rational numbers like 1/2, 3/4
• Irrational numbers such as \( \sqrt{2} \) and \( \pi \)

A real number is denoted by \( \mathbb{R} \).
In the context of upper bounds, real numbers are used to define limits or constraints on subsets of numbers. Understanding upper bounds using real numbers helps in defining the limits of a sequence or function, contributing to solving equations or inequalities more confidently.

Subset

A subset is a set where all elements of the first set are contained within a second set. The notation \( A \subseteq B \) indicates that every element of set \( A \) is also an element of set \( B \).
Subsets play a crucial role in analysis as they offer a framework for understanding collections of objects or numbers within a larger set. For example, if we consider the set of real numbers \( \mathbb{R} \), a subset \( A \) could be \( \{1, 2, 3\} \).
In real analysis, identifying subsets allows us to examine properties of these smaller collections in more manageable ways, such as determining the existence of upper or lower bounds.
Recognizing how subsets work aids in exploring how different sets relate to each other from within a larger context.

Inequality

An inequality is a mathematical statement that compares two values, showing how they are not equal and indicating the relationship between them. The basic symbols used include:

• \( > \) Greater than
• \( < \) Less than
• \( \geq \) Greater than or equal to
• \( \leq \) Less than or equal to

Inequalities are used to determine the upper and lower bounds of sets. In the context of real numbers, inequalities assist in identifying whether a number acts as a boundary or limit for other numbers in a set.
For example, with the subset \( A = \{1, 2, 3\} \) and a supposed upper bound \( K^* = 2 \), we check the condition \( x \leq 2 \).
However, since 3 is an element of \( A \) and \( 3 > 2 \), \( K^* = 2 \) is not an upper bound.
This illustrates how inequalities help verify whether a number qualifies as an upper bound by checking against specific conditions.