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Discrete mathematics is an area of study focused on mathematical structures that are fundamentally discrete rather than continuous. This means that the values dealt within discrete mathematics do not vary smoothly, but have distinct, separated values. In the context of our exercise, the sequence of numbers is an example of such a discrete system. Each number in the sequence is distinct from the next, related through a specific formula or property, reflecting the stepwise nature of discrete phenomena. Discrete mathematics encompasses a wide range of topics, including set theory, combinatorics, graph theory, and logic, all of which provide foundational tools for computer science and algorithms.

Understanding discrete mathematics involves mastering the art of creating and manipulating discrete elements, such as sequences or sets, to solve problems. In the context of increasing sequences, it embodies the idea of crafting a progression of numbers, each following a pattern or rule, which is precisely what the exercise requires us to construct.

The concept of natural numbers is instrumental in our understanding of sequences and series. The natural numbers, commonly denoted by \(\mathbb{N}\), are the set of numbers used for counting and ordering. They are the numbers \(1, 2, 3, \ldots\) and so on ad infinitum. A key characteristic of natural numbers is that they are discrete, meaning there is a well-defined next number (successor), but there is no number between any two consecutive natural numbers.

For the exercise in question, the natural numbers serve as the indexing set for the sequence. The indexes \(n\) which signify the position of terms within the sequence \(a_n\), are themselves natural numbers. This is why it's important to understand how to construct sequences indexed by natural numbers and the properties that arise from doing so, such as the assurance of being able to continually generate discrete, new terms that maintain the strictly increasing nature of the sequence.

The harmonic series is a famous infinite series, which is defined as the sum of the reciprocals of the natural numbers. Mathematically, it is represented as \(\sum_{n=1}^{\infty} \frac{1}{n}\). One interesting property of the harmonic series is that it diverges, meaning it increases without bound as more terms are added. However, the harmonic series grows very slowly, which becomes highly relevant in our exercise when designing a sequence that grows but always remains below a certain limit, like 100.

In constructing our strictly increasing sequence, we've essentially tapped into the nature of the harmonic series by adding decreasing fractions to each consecutive term. These fractions are tiny enough to ensure our sequence doesn't surpass our ceiling of 100. The harmonic series serves as a conceptual anchor to understand how a sequence can keep growing indefinitely in a controlled manner without limit, yet still remain finite when looking at any specific term in the series.

Proof construction is a fundamental concept in mathematics that involves formulating arguments to establish the truth of mathematical statements. Proofs can use direct methods, contradiction, induction among others, and aim to ensure that conclusions follow logically and incontrovertibly from premises.

In the context of our exercise, we have constructed a proof by demonstrating the existence of a strictly increasing sequence that remains below 100. This was done by specifying a clear starting point and a rule for progression, then verifying that the conditions hold true. Proof construction in this case was not about showing that a number will eventually exceed 100, but rather that it is possible to approach it indefinitely while remaining strictly increasing and below this threshold. It showcases how mathematical reasoning is employed to offer insight and certainty in problem-solving.