apply Zlog_sup_seq. apply Mset_l. apply Zle_lt_trans with (w - 1).
rewrite <- Zlog_sup_pow with (i := w - 1). apply Zlog_sup_seq. apply Mset_r. cut (0 < w). omega.
exact lt_0_w. omega. Qed.

Lemma tex : forall M : Mset, in_Z_ w (w - Zlog_sup M - 1 / M).
intro M. unfold in_Z_. elim (Zle_lt_or_eq 1 M). intro H0. replace (1 / M) with 0. split.