In the (arXiv) paper, Exhaustive search of convex pentagons which tile the plane by Michael Rao, on page 4 under the proof of Lemma 2, it is said that:

"… We keep a connected component $H_d'$ of $H_{d}$ such that $S(H_{d}')$ contains the centre of the square. Then one can construct by compactness an infinite graph in which every vertex is an interior vertex, and with $W(G') \subseteq W(T)-\{v\}$. There are three cases: either $G'$ corresponds to a tiling of the plane, of a half plane or of a stripe. In all cases, one can construct a tiling of the place without vertex of vector type $v$, and no new vector type."

Searching "construct by compactness" yields very few results, and each result is equally vague about what this means. It seems as if there is some "well known" procedure which the author is alluding to, but I have no idea what it is. Any suggestions will be appreciated.

I'm happy to update this question with pertinent details about the preceding content of the paper, but I'm not sure which details will be relevant, so for now I want to keep this question brief.