Full Idea
Compactness is a corollary of soundness and completeness. If Γ is not satisfiable, then, by completeness, Γ is not consistent. But the deductions contain only finite premises. So a finite subset shows the inconsistency.
Gist of Idea
Compactness is derived from soundness and completeness
Source
Stewart Shapiro (Foundations without Foundationalism [1991], 4.1)
Book Reference
Shapiro,Stewart: 'Foundations without Foundationalism' [OUP 1991], p.79
A Reaction
[this is abbreviated, but a proof of compactness] Since all worthwhile logics are sound, this effectively means that completeness entails compactness.