For the chromatic number $\chi(G)$ of a simple, undirected graph, there is a "compactness" theorem by Erdős and De Bruijn stating that if an infinite graph $G$ has finite chromatic number, then there is a finite subgraph $G_0\subseteq G$ such that $\chi(G_0) = \chi(G)$.

A converse statement would be

$(\text{S})$ Let $k>0$ be an integer. Whenever a graph $G$ has the property that every finite subgraph $G_0$ of $G$ can be colored with $k$ colors, then $G$ can be colored with $k$ colors.

Is $(\text{S})$ true?