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(**           Convergence properties of sequences                *)
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Definition Un_decreasing (Un:nat -> R) : Prop :=
  forall n:nat, Un (S n) <= Un n.Definition opp_seq (Un:nat -> R) (n:nat) : R := - Un n.Definition has_ub (Un:nat -> R) : Prop := bound (EUn Un).Definition has_lb (Un:nat -> R) : Prop := bound (EUn (opp_seq Un)).

Lemma decreasing_growing : forall Un:nat -> R, Un_decreasing Un -> Un_growing (opp_seq Un).