Infinite-dimensional vector spaces and their bases!

I was working on a problem earlier today and I didn't know the following result:

Let S be a subset of an infinite-dimensional vector space V. Then S is a basis for V if and only if for each nonzero vector v in V, there exists unique vectors u1,u2,...,un in S and unique nonzero scalars c1,c2,...cn, such that v = (c1)u1 + (c2)u2 + ... + (cn)un.

I don't "see" how this can be true? For example, lets say I take the vector space of infinite-tuples so x = (x1, x2, ...). How is it that I can write this as a linear combination of FINITE number of elements of S (a basis of this vector space)? It just seems that I'd require an infinite number of elements of S to do so. Can anyone help me understand this?

Thanks!