[foobar__]

This falls short of the original goal: While your method shows that for every point, you can find a tiling large enough that covers it, the original question was to find a single tiling covering everything.

Finding progressively larger, finite tilings is not the same as having a single infinite tiling, just like finding larger and larger natural numbers is not the same as having a single number larger than all natural numbers (which wouldn't be a natural number).

König's lemma implies that for tilings both statements are in fact equivalent.

[ot]

> like finding larger and larger natural numbers is not the same as having a single number larger than all natural numbers

Very nice analogy!

> König's lemma implies that for tilings both statements are in fact equivalent.

Exactly, and instead of working by shifting the tiling (which would produce a sequence of incompatible tilings) it works by finding a sequence of finite tilings where each is a subset of the next, so it makes sense to take the union of the sequence.

[nightcracker]

With the caveat that if your tiling is inductive in the sense that the tiling of n+1 is an extension of n, a series of finite tilings will tile the plane.