Existence of subsequences of sequences of real numbers under certain conditions I want to show that if $n > srp$, then any sequence of $n$ real numbers must contain either
a strictly increasing subsequence of length greater than $s$, a strictly decreasing subsequence of length greater than $r$, or a constant subsequence of length greater than $p$.
I thought I can prove this by assuming that two of the conditions do NOT hold, and forcing the third as a consequence. We would have to do this thrice to complete the proof. But I am stuck after assuming the hypothesis. Is there a more direct way to prove this or am I on the right track?
 A: Contradiction is a good approach, but we don't need to overcomplicate things with separate cases. We only need to show that one must condition hold, not that they all must, so we could suppose that none of them do and then demonstrate a single contradiction. Instead, we adapt a proof of the Erdős–Szekeres theorem given by Seidenberg and argue the contrapositive.
Let $m\in\mathbb{N}$ and let $(x_k)_{k\leq m}$ be any real sequence with no strictly increasing, strictly decreasing, or constant subsequences of length greater than $s$, $r$, or $p$ respectively. For each $1\leq k\leq m$, let $i_k\in\{1\ldots, s\}$ represent the length of the longest strictly increasing subsequence of $(x_l)_{l\leq k}$ ending with $x_k$. Define $d_k$ and $c_k$ analogously for decreasing and constant subsequences.
Now, define $f:\{1,\ldots ,m\}\rightarrow \{1,\ldots ,s\}\times\{1,\ldots, r\}\times\{1,\ldots,p\}$ by $f(k)=(i_k,d_k,c_k)$. For any distinct $1\leq \alpha<\beta\leq n$, $x_\beta$ is either greater than, less than, or equal to $x_\alpha$, so there is either a strictly increasing, strictly decreasing, or constant subsequence of $(x_k)_{k\leq\beta}$ ending with $x_\beta$ of length greater than $i_\alpha$, $d_\alpha$, or $c_\alpha$. Thus, $f(\alpha)\neq f(\beta)$, so $f$ is injective and $m\leq srp$. Thus, any real sequence of length $n>srp$ must contain some desired subsequence.
