Question Number 100769 by Rio Michael last updated on 28/Jun/20

Consider the sequences (u_n ) and (v_n ) defined by { ((u_0 = 1)),((u_(n+1) = ((2u_n v_n )/(u_n + v_n )))) :} and { ((v_0 = 2)),((v_(n+1) = ((u_n + v_n )/2))) :} ∀ n∈ N (1) Show that (u_n ) and (v_n ) are strictly positive also Show that (u_n ) and (v_n ) are of opposite sense of variation. (2) let w_n = v_n −u_n show that 0 ≤ w_(n+1) ≤ (1/2)w_n (3) Prove by induction that 0 ≤ w_n ≤ (1/2^n )