It is known that for a given matrix A of rank r, and a set D of positive diagonal matrices, Ⅱ()1222supⅡ()Ⅱ(min())iiWDWAWAσ+−+∈=, in which is a submatrix of a formed with r=rank (A) rows of A, such that has full row rank r. In many practical applications this value is too large to be used.In this paper we consider the case that both A and ()iA()iAWD∈ are fixed with W severely stiff. We show that in this case the weighted pseudoinverse 112()WAW+ is close to a multi-level constrained weighted pseudoinverse therefore 11222Ⅱ()ⅡWAW+is uniformly bounded. We also prove that in this case the solution set of the stiffly weighted least squares problem is close to that of corresponding multi-level constrained least squares problem.