Projects & Research A Radical Approach to Real Analysis Macalester College


Chapter 1: Crises in Mathematics: Fourier's Series

Chapter 2: Infinite Summations

2.1 Avoiding Infinite Series

2.2 The Geometric Series

2.3 Calculating Pi

2.4 The Harmonic Series

Euler's Constant

Proving Convergence

> The Nested Interval Principle

2.5 Taylor Series

2.6 Emerging Doubts

2.4 The Harmonic Series (continued)

The Nested Interval Principle

If we look closely at our situation, the solution seems too obvious to need special attention. We have created a sequence of nested intervals.

Figure 2.3 Nested Intervals

Surely there must be some number contained in the intersection of all of them.

It was not until late in the nineteenth century that mathematicians realized that this “obvious fact” was in fact a subtle point that really defined the nature of the real number line. It is the crucial property that distinguishes the set of all real numbers from the set of rational numbers.

The Nested Interval Principle

Given an increasing sequence and a decreasing sequence such that is always larger than , but the difference between and can be made arbitarily small by taking n sufficiently large, there is exactly one real number that is greater than or equal to every and less than or equal to every .

The conclusion that there is at least one such number is something we cannot prove, not without making some other assumption that is equivalent to the nested interval principle. We have reached one of the foundational assumptions on which a careful and rigorous treatment of calculus can be built. It took mathematicians a long time to realize this. In the early 1800s, the nested interval principle was used as if it was too obvious to bother justifying or even stating very carefully. What it guarantees is that the real number line has no holes in it. If two sequences are approaching each other from different directions, then where they ``meet'' there is always some number.

This principle will play an important role in future chapters when we enter the nineteenth century and begin to grapple with questions of continuity and convergence. It will be our primary tool for showing that a desired number actually exists even when we do not know what it is.

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