Sounds like it should be b is in THE span OF (the column vectors of) A. The proof would just be knowing the definition of span, plus multiplying A times x and writing it out in terms of the components of x.
The "range" of A, the set of all vector of the form Av for some v in the domain of A, is sometimes called the "span" of A. Since we can take, in succession, v= <1, 0, 0, ..., 0>, v= <0, 1, 0, ..., 0>, v= <0, 0, 1, ..., 0>, to v= <0, 0, 0, ..., 1> and applying A to each of those gives a column of A, it can be shown that the columns of A span the range of A. And, if A is invertible, form a basis for it.