whence we obtained (3.4).
Since f is ≤ p -continuous, we have that f ( x ∗ ) is the least upper bound of ( f n ( x 0 ) ) n ∈ N. Whence we immediately obtain that f ( x ∗ ) = x ∗ and that x ∗ ∈ ↑ x 0. □.
whence for any fixed, after substitution of,, we obtain (2.5).
Since the last inequality is true for any η > 0, we obtain Δ X,β (d/r) = 0, whence ε β (X) ≥ d/r.
By Theorem 9 we obtain v h ⊑ s p w. Whence we deduce that h ≤ f w and, hence, that f w ∈ Ω ( h ) or, equivalently, that f T ∈ Ω ( h ).
If m ( A t 0, x, y ) ≥ α − t 0, then we can assume without loss of generality that e t 0 ⊂ e α ⊂ e t 0 ∪ A t 0, x, y, whence we get the equality m ( supp x χ e α ∩ supp y χ e α ) = 0. Proceeding analogously as in Case 2.2.1.1, we obtain I φ, ω ( x + y 2 ) < 1.
Since Φ z ( u ) ⊑ s p u we obtain q B ( Φ z ( u ), u ) = 0. Hence, by assertion (1) in statement of Lemma 8, wefind that q B ( w, u ) = 0. Whence we deduce that l ⪯ ( w, u ) = ∞ and, hence, that w ⊑ s p u.
We obtain this as follows.
