While proving that every vector space has a basis, why are only finite linear combinations used in the proof?

The definition of linear independence is that any finite linear relation is trivial.

Vector spaces in general do not have any concept of an infinite sum at all. For those vector space where the usual concept of an infinite sum of reals can be generalized, one may speak of a different kind of span/basis where one allows infinite linear combinations in addition to finite ones. That gives rise to a separate concept, different from the usual kind of linear-combinations basis.

When one needs to distinguish between the different notions of basis, an ordinary basis that works by finite linear combinations is called a "Hamel basis" or "algebraic basis", and one that needs infinite linear combinations to span everything is called a "Schauder basis" (though strictly speaking the latter name implies some additional conditions).

A vector space has no notion of convergence or order. Since these are required for assigning meaning to an infinite sum, we are restricted to finite sums.

In the study of normed linear spaces, the notion of Schauder basis is an object of study.