Proof using.
intros.
eapply inf_often_monotonic with (P:=invariant /\_ P).
- intros.
unfold and_tl in *; firstorder.
- eapply cumul_inf_often_always; eauto.
Qed.

Lemma continuously_monotonic :
forall (P Q : infseq T -> Prop),
(forall s, P s -> Q s) ->
forall s, continuously P s -> continuously Q s.