Let A_r = a_(i j) be a sequence of N symmetric matrices of increasing order with i, j = 1, 2, ..., r and r = 1, 2, ..., N. Let λ_k(A_r) be the kth eigenvalue of A_r for k = 1, 2, ..., r, where the ordering is given by λ_1(A_r)>=λ_2(A_r)>=...>=λ_r(A_r). Then it follows that λ_(k + 1)(A_(i + 1))<=λ_k(A_i)<=λ_k(A_(i + 1)).