Combinatorial techniques, methods, and ideas in (“undergraduate”) real analysis

Question

This question is dual to Probabilistic techniques, methods, and ideas in ("undergraduate") real analysis: I would like to collect some examples of combinatorial arguments to undergraduate (so to say) real analysis (that is, to calculus problems).

Answer 1

It seems the following.

Proposition. Each sequence of real numbers contains a monotonic subsequence.

Proof. Let $\{a_n:n\in\Bbb N\}$ be a sequence of real numbers. Consider the following 2-coloring of the edges of the graph $K_{\Bbb N}$ (that is the graph whose vertices are natural numbers and each two different vertices are joined by an edge). Let $m<n$ be natural numbers. Color an edge which joins vertices $m$ and $n$ red if $a_m\le a_n$ and green otherwise. Infinite Ramsey Theorem implies that there exists an infinite subset $M$ of $\Bbb N$ such that all edges which join vertices of $M$ have the same color. Then the sequence $\{a_n:n\in M\}$ is a monotonic subsequence of the sequence $\{a_n:n\in\Bbb N\}$.$\square$