18.3.3. Conclusion on Divisibility

We want to show that if you take any sequence of n + 1 distinct real numbers, then there always exists a subsequence of length n + 1 that is either strictly increasing or strictly decreasing.

The main idea revolves around finding subsequences in a given sequence of distinct real numbers. A strictly increasing subsequence is a sequence of elements where each element is greater than the one before it. Conversely, a strictly decreasing subsequence is where each element is less than the one before it. The challenge is to prove that no matter how we choose these n + 1 numbers, one of these subsequences (either increasing or decreasing) will always exist.

Think of a group of friends whose heights vary. If you take a group of 8 friends (which is n + 1, where n = 7) and you want to find a way to stand them in a line such that their heights are either all increasing or all decreasing, you will always be able to arrange at least some of them in this manner because there are just enough distinct heights that force this situation due to the nature of how numbers work.

A strictly increasing sequence is of the form (a₁, a₂, …) where a₁ < a₂ < a₃ < ... < aₘ, while a strictly decreasing sequence is of the form (a₁, a₂, …) where a₁ > a₂ > a₃ > ... > aₘ.

Here, we clarify what strictly increasing and strictly decreasing sequences are. An increasing sequence means that as we move through the sequence from the first element to the last, each next element is larger than the previous. On the other hand, in a decreasing sequence, each next element is smaller than the previous one. Understanding these definitions is crucial for the following steps in the proof.

Consider a set of stairs. If you keep stepping up, you're moving in an increasing manner. If you're stepping down, that's like a decreasing sequence. Just like you can't stand on the same step twice if you're going up or down, the elements in these mathematical sequences must be distinct.

To prove our claim, we use the pigeonhole principle along with proof by contradiction. Assume that for every number in the sequence, the length of the longest increasing subsequence is at most n.

The pigeonhole principle is a common inference tool which states that if you have more items than containers, at least one container must contain more than one item. In our case, if we assume that every sequence can hold only a certain amount of longest subsequences, we eventually get a contradiction when we try to pair these subsequences. Since we have more distinct elements than the capacity of subsequences, we conclude that at least one subsequence must exceed the assumed maximum length.

Imagine you have 24 pairs of socks but only 23 drawers to store them in. According to the pigeonhole principle, at least one drawer must contain more than one pair of socks. This represents how we cannot have more sequences of length at most n than there are unique numbers available.

Assume that there exists some numbers that contradict the maximum length assumption. If m is the number of pairs found, we can rearrange them such that we have a longer subsequence.

By assuming the opposite of the claim (that no subsequence can be longer than n), we show that this leads to an impossibility in how the pairs are structured. When we rearrange them under the earlier defined relations of increasing or decreasing subsequences, we eventually find that at least one subsequence must necessarily break the initial assumption, therefore confirming the claim.

Think of a group of students trying to arrange their chairs in a line. If you have more students than chairs, a couple of students must step out of line to fit into the arrangement without sitting in the same chair. Thus, if each divided group must have some overlapping seating, it leads you to realize that at least one student must reappear in a different sequence.

The existence of either an increasing or decreasing subsequence of length n + 1 is guaranteed by the properties laid out thus far.

In conclusion, by utilizing definitions of strictly increasing and decreasing sequences along with the pigeonhole principle, we assert that any selection of distinct real numbers will inherently contain either an increasing or decreasing arrangement of length n + 1. This universality in behavior reinforces the fundamental structures within number theory and sequences.

This conclusion parallels common experiences. For instance, if you have a set of unique colors and you try to arrange them in a sequence without repeating, eventually, you'll find a similar pattern of color arrangements that either transition smoothly from light to dark or vice-versa, due to the nature of distinct yet relatable shades.

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