Full Idea
The logic of truth-functions is compact, which means that sequents with infinitely many formulae on the left introduce nothing new. Hence we can confine our attention to finite sequents.
Gist of Idea
Compactness means an infinity of sequents on the left will add nothing new
Source
David Bostock (Intermediate Logic [1997], 5.5)
Book Reference
Bostock,David: 'Intermediate Logic' [OUP 1997], p.217
A Reaction
This makes it clear why compactness is a limitation in logic. If you want the logic to be unlimited in scope, it isn't; it only proves things from finite numbers of sequents. This makes it easier to prove completeness for the system.
Related Idea
Idea 13630 Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures [Shapiro]