Every enumeration of a cofinite set of integers has in-order 3-APs from every start

Let $t:\mathbb{N}\to\mathbb{Z}$ be any sequence that takes every integer outside some finite set $F$ as a value. Then for every index $i$ there are indices $i<j<l$ with

i.e. $t_i,t_j,t_l$ is a nonconstant monotone $3$-term arithmetic progression in index order, starting at any prescribed position. (Proof idea: choose $x$ far from $F$ and from $t_0,\dots,t_i$; the positions of $t_i+2^n(x-t_i)$, $n\ge0$, cannot decrease forever.) In connection with Erdős Problem 195, this shows that every enumeration of a cofinite subset of $\mathbb{Z}$ has in-order $3$-APs starting arbitrarily late.

Formalization note: this is a purely combinatorial statement about integer sequences, phrased with t : ℕ → ℤ (or a chain of lists P n with P (n+1) = P n ++ [w n]); "cofinite" is expressed as existence of a finite set F : Finset ℤ outside of which every integer occurs.