A sequence of consecutive line segments in $\mathbb{R}^2$ is called a Cap if their slopes are monotonically decreasing, and a Cup if their slopes are monotonically increasing. Let $f(s,t)$ denote the smallest number for which any collection of $f(s,t)$ points in general position either contains a Cap of length $s$ or a Cup of length $t$.
I already proved that $f(s,t) \leq f(s−1,t) + f(s,t−1) + 1 \text{ for all } s,t \geq 4$ $(*)$.
To prove the inequality in the title, I tried induction on $(s+t)$, applying it to $f(s−1,t) \text{ and } f(s,t−1)$ in $(*)$. Yet all I got is $f(s,t) \leq {s+t-2 \choose {s-2}} + 3$ and can't find any way to reduce the last term to $1$. Did I do something wrong?