Lifting chains of prime ideals

We give an elementary proof that for a ring homomorphism A -> B, satisfying the property that every ideal in A is contracted from B, the following property holds: for every chain of prime ideals p_0 \subset … \subset p_r in A there exists a chain of prime ideals q_0 \subset … \subset q_r in B such that q_i \cap A = p_i.