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 )