Guaranteed hit on a submarine with unknown start and speed

A submarine sits at an unknown integer position on a line at time $0$ and each turn moves by the same unknown integer amount (possibly negative or zero). Once per turn you may fire at one integer point. Describe a strategy that is certain to hit the submarine after finitely many turns.

Enumerate all (start, step) pairs. The pairs $(a,b)\in\mathbb Z^2$ are countable, so list them as $(a_0,b_0),(a_1,b_1),\dots$. At turn $t$ fire at $a_t+b_t t$, where the submarine would be if its parameters were the $t$-th pair. The true pair appears at some finite index $T$, and that shot hits.