If n+1 or more objects are placed into n boxes, then there is at least one box containing two or more objects. In other words, if A and B  are two sets such that |A| > |B|, then there is no one-to-one function from A to B.

Theorem 1: Let n be a positive integer. Every sequence of n² + 1 distinct real numbers contains a subsequence of length n+1 that is either increasing or decreasing.

Proof. For example consider the sequence (20, 10, 9, 7, 11, 2, 21, 1, 20, 31) of 10 = 3² + 1 numbers. This sequence contains an increasing subsequence of length 4 = 3 + 1, namely (10, 11, 21, 31).

The proof of this theorem is by contradiction, and uses the pigeon hole principle.

Let (a₁, a₂,…..,a_n²+1) be an arbitrary sequence of n² + 1 distinct real numbers. For each i with 1 ≤ i ≤ n² + 1, let inc_i denote the length of the longest increasing subsequence that starts at a_i, and let dec_i denote the length of the longest decreasing subsequence that starts at a_i
Using this notation, the claim in the theorem can be formulated as follows:

There is an index i such that inc_i ≥ n + 1 or dec_i ≥ n + 1.

We will prove the claim by contradiction. So we assume that inc_i ≤ n and dec_i ≤ n for all i with 1 ≤ i ≤ n² + 1.
Consider a set

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