Definition of a span Let $S \subset V$ with $V$ a vector space.
I managed to prove that Span(S) is the smallest subspace of V containing S. The end of the exercise says: "We conclude that $Span(S)$ consists of all linear combinations of finite sets of elements of S.
Why finite sets and not infinite sets?
 A: You can't really talk about an infinite sum without brining in the notion of convergence.  You need some way of saying that points are getting close to each other.  In Euclidean space, $\mathbb R^n$, you can do this, but in other vector spaces such a notion of closeness may not be defined.
A: The assertion

(A) all linear combinations of finite sets of elements of $S$

means

(B) all finite linear combinations of elements of $S$

So, the assertion

(C) all linear combinations of infinite sets of elements of $S$

would mean

(D) all infinite linear combinations of elements of $S$

But, in the context of vector spaces,
(E) the definition of linear combination involves only finitely many vectors (because "purely algebraic operations do not permit us to introduce an adequate notion of convergence").
From this point of view, (D) is meaningless and this is why the original assertion considers "finite sets".
Remark: note that, in view of (E), (B) is redundant. Thus the original assertion could be written as all linear combinations of elements of $S$.
