Exercise. In contrast to the proof we give below, which has...

Exercise. In contrast to the proof we give below, which has certain pedagogical motivations, construct a proof using the idea in the remark above. For example, consider the intervals of the form $[q|b|,(q+1)|b|)$ where $|b|$ is the absolute value of $b$ and $q$ ranges over all the integers.

Key Concepts

Partitioning of the Real Number Line

This concept involves dividing the whole set of real numbers into a sequence of disjoint intervals of equal length. By partitioning the real number line, one can isolate and focus on specific segments of interest and apply uniform reasoning across all intervals. This strategy is used in proofs to simplify problems by reducing them to studying the behavior on each interval, making it easier to apply global arguments through local inspection.

Pigeonhole Principle

The pigeonhole principle is a fundamental counting argument that asserts if more objects are distributed among fewer containers, then at least one container must contain more than one object. In many mathematical proofs, this principle is used after partitioning a set into intervals or classes to conclude that some interval or class must contain multiple elements, which can then be leveraged to prove the existence or uniqueness of a particular structure or value.

Properties of Absolute Value

The absolute value function measures the distance of a number from zero, thereby providing a notion of size irrespective of sign. This is crucial when working with intervals defined by multiples of a quantity, as it ensures that the intervals maintain a consistent length and translate symmetrically along the number line. These properties are essential when constructing proofs that rely on the uniformity of interval lengths and handling both positive and negative values uniformly.

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