Indeed, if not, then there would exist an ϵ > 0 and subsequences { Z m ( t ) } and { Z n ( t ) } of { Z n } such that n ( t ) is the minimal in the sense that n ( t ) > m ( t ) ≥ t and ρ k ( Z m ( t ), Z n ( t ) ) > ϵ.
If for every $e$ there would exist a number $n$ such that ${\bf PA}\vdash T e,e,n) \vee \forall y \neg T e,e,y $, then by checking whether $T e,e,n)$ holds it would be decided whether a program $e$ terminates on input $e$.
(31) If (31) were not true, then there would exist a positive integer (n_{1}) such that (0< I t)
If (40) were not true, then there would exist a sequence of initial values (Y_{n}=(S_{n},I_{n},R_{n},U_{n},V_{n})inmathbb{R}_^{5}) ((n=1,2,ldotsuchsuch that liminf_{trightarrowinfty}I t, Y_{n})< frac{varepsilon}{n^{2}}.
Remark 3.1 Let g and F be as in Theorem 2.2 and let x 0 ∈ A. If the maps g and x ↦ F ( x, x ) are continuous at point x 0, then G is continuous on A. Namely, if G was not continuous, then there would exist an integer C and a sequence { x k } k = 0 ∞ such that lim k → ∞ x k = 0 and ∥ G ( x k ) ∥ > 1 C for k ≥ 0. Write F ˜ ( x, x ) = F ( x, x ) 1 − ξ η, x ∈ A. Let t > C ( 2 F ˜ ( x 0, x 0 ) + 1 ).
If was discontinuous at a point, then there would exist sequences and in such that and, are convergent,, and with.
for every t ∈ [0,T m ). If this was not the case, then there would exist a time t1 such that t 1 = min t ∈ ( 0, T m ) : I ( u ( t ) ) = 0 > 0. (4.20..
If not, there would exist (uin partial_{K}Omega^{2}) and (lambda>0) such that (u=S u+lambda w).
One can entertain the existence of unicorns and their necessary features (that necessarily if there were unicorns, there would exist single-horned beasts) without believing that there are unicorns.
