Consider the iterative systems u(k+1) = u(k) and v(k+1) = u v(k) where || > |u|. Prove

Question:

Consider the iterative systems u(k+1) = λu(k) and v(k+1) = u v(k) where |λ| > |u|. Prove that, for any nonzero initial data u(0) = a ≠ 0. v(0) = b ≠ 0, the solution to the first is eventually larger (in modulus) than that of the second: |u(k))| > ||v(k)|, for k >> 0.