Problem 8 - Advanced Calculus

Question

Show that the intersection \(\bigcap_{1}^{x} I_{n}\) of the nested sequence of intervals \(\left\\{I_{n}\right\\}\) is empty in the following cases: (a) \(I_{n}\) is the open interval \(0

Step-by-Step Solution

Answer

For both cases (a) and (b), there is no real number that is common to all intervals \(I_n\). Hence, the intersection \(\bigcap_{1}^{x} I_{n}\) is empty.

Step 1: Case (a) Solution

In case (a), the intervals \(I_{n}\) are represented as the open interval \((0, 1/n)\), i.e., the set of all numbers greater than 0 and less than 1/n. As n increases, the upper limit of these open intervals decreases, which results in progressively smaller intervals. The intersection of all these intervals \(\bigcap_{1}^{x} I_{n}\) would consider only those values which are common to all intervals. As the intervals keep getting smaller with increasing n, there is no real number that is common to all, which means that the intersection is empty.

Step 2: Case (b) Solution

In case (b), the intervals \(I_{n}\) are represented as the closed interval \([n, \infty)\), i.e., the set of all numbers greater than or equal to n. As n increases, the lower limit of these closed intervals increases, which results in progressively larger intervals. The intersection of all these intervals \(\bigcap_{1}^{x} I_{n}\) would consider only those values which are common to all intervals. Given that the lower limit keeps increasing, there is again no real number that is common to all. Therefore, the intersection is also empty in this case.

Key Concepts

Empty Intersection

The concept of an empty intersection revolves around the idea of finding a common element among a series of sets, where no single element is shared across every set. In mathematical terms, an intersection refers to the set of elements that are present in all considered sets. When we talk about an empty intersection, it means there is no element that exists in all the sets, hence the intersection results in an "empty set," denoted as \( \emptyset \). This concept can be observed in nested sequences of intervals, where intervals shrink or shift, resulting in no overlap for a common element as illustrated in the original exercise with different cases of open and closed intervals.

Open Interval

An open interval in mathematics is a set of real numbers that includes all numbers between two endpoints but not the endpoints themselves. It is denoted as \((a, b)\), where \(a\) and \(b\) are the endpoints. Open intervals have some distinctive properties:

• The interval does not include the endpoints; that is, \(a < x < b\).
• They are typically used in calculus to explore limits and continuity without boundary constraints.
• Open intervals are part of the real number line and represent a continuous set of numbers between \(a\) and \(b\).

In case (a) of the exercise, as \(n\) increases, the open interval \((0, 1/n)\) decreases in width making it impossible for any real number to exist in all intervals simultaneously, thereby resulting in an empty intersection.

Closed Interval

Closed intervals are another important concept in understanding the structure of real numbers over a set range. A closed interval includes its endpoints, denoted as \([a, b]\). Within this interval:

• All numbers between \(a\) and \(b\) are included, as well as \(a\) and \(b\) themselves; that is, \(a \leq x \leq b\).
• Closed intervals are particularly useful in problems and proofs, where endpoint values are essential.
  
• They form a complete bound or closure within the number line.

In the scenario described in case (b), the intervals \([n, \infty)\) keep expanding as \(n\) increases, pushing the start of the interval further along the number line. As no single real number can keep up with the increasing starting point across all intervals, their intersection turns out to be empty.

Real Numbers

Real numbers form the core of many mathematical concepts, spanning everything from simple arithmetic to complex analysis. They include all rational and irrational numbers and can be thought of as points on an infinite line, known as the real number line. Key features of real numbers include:

• They encompass whole numbers, fractions, and decimal numbers, both terminating and repeating.
• The set of real numbers is continuous with no breaks or gaps. This completeness implies every point on the number line corresponds to a real number.
• Real numbers are ordered; meaning, for any two real numbers, you can determine which is larger or smaller.

In the context of our nested intervals, real numbers highlight the lack of a single value that exists within all \(I_n\) intervals as they shift or shrink. The property of order is crucial, as it demonstrates why these intervals lack a common real number leading to an empty intersection.