Well, take any element x in K(A): since this is a vector space over K generated by A (perhaps not finite dimensional, though) , there exist elements y_1,...,y_n in K(A) s.t. x is K-linear combination of y_1,...,y_n.
Now, each element y_i is a K-lin. comb. of a finite number of elements from A since, as pointed out above, K(A) is generated as K-vec. space by A ==> in y_1,...,y_n intervene only a finite number of elements of A ==>
the element x is a K-lin. com. of a finite number of elements of A, say a set M ==> x belongs to K(M), for M a finites subset of A.