lemma segment_preserves_uparrow:
∀C:ordered_set.∀l,u:C.∀a:sequence {[l,u]}.∀x,h.
- â\8c\8an,\fst (a n)â\8c\8b â\86\91 x â\86\92 a â\86\91 â\8c©x,hâ\8cª.
+ â\8c\8an,\fst (a n)â\8c\8b â\86\91 x â\86\92 a â\86\91 â\89ªx,hâ\89«.
intros; cases H (Ha Hx); split [apply Ha] cases Hx;
split; [apply H1] intros;
cases (H2 (\fst y)); [2: apply H3;] exists [apply w] assumption;
lemma segment_preserves_downarrow:
∀C:ordered_set.∀l,u:C.∀a:sequence {[l,u]}.∀x,h.
- â\8c\8an,\fst (a n)â\8c\8b â\86\93 x â\86\92 a â\86\93 â\8c©x,hâ\8cª.
+ â\8c\8an,\fst (a n)â\8c\8b â\86\93 x â\86\92 a â\86\93 â\89ªx,hâ\89«.
intros; cases H (Ha Hx); split [apply Ha] cases Hx;
split; [apply H1] intros;
cases (H2 (\fst y));[2:apply H3]; exists [apply w] assumption;
∀C:ordered_uniform_space.property_sigma C →
∀l,u:C.exhaustive {[l,u]} →
∀a:sequence {[l,u]}.∀x:C. ⌊n,\fst (a n)⌋ ↑ x →
- xâ\88\88[l,u] â\88§ â\88\80h:x â\88\88 [l,u].a uniform_converges â\8c©x,hâ\8cª.
+ xâ\88\88[l,u] â\88§ â\88\80h:x â\88\88 [l,u].a uniform_converges â\89ªx,hâ\89«.
intros; cases H2 (Ha Hx); clear H2; cases Hx; split;
[1: split;
[1: apply (supremum_is_upper_bound C ?? Hx u);
|2: apply (le_transitive ? ??? ? (H2 O));
apply (segment_lowerbound ?l u);]
|2: intros;
- lapply (uparrow_upperlocated ? a â\8c©x,hâ\8cª) as Ha1;
+ lapply (uparrow_upperlocated ? a â\89ªx,hâ\89«) as Ha1;
[2: apply segment_preserves_uparrow;split; assumption;]
- lapply (segment_preserves_supremum ? l u a â\8c©?,hâ\8cª) as Ha2;
+ lapply (segment_preserves_supremum ? l u a â\89ª?,hâ\89«) as Ha2;
[2:split; assumption]; cases Ha2; clear Ha2;
cases (H1 a a); lapply (H6 H4 Ha1) as HaC;
lapply (segment_cauchy ? l u ? HaC) as Ha;
∀C:ordered_uniform_space.property_sigma C →
∀l,u:C.exhaustive {[l,u]} →
∀a:sequence {[l,u]}.∀x:C. ⌊n,\fst (a n)⌋ ↓ x →
- xâ\88\88[l,u] â\88§ â\88\80h:x â\88\88 [l,u].a uniform_converges â\8c©x,hâ\8cª.
+ xâ\88\88[l,u] â\88§ â\88\80h:x â\88\88 [l,u].a uniform_converges â\89ªx,hâ\89«.
intros; cases H2 (Ha Hx); clear H2; cases Hx; split;
[1: split;
[2: apply (infimum_is_lower_bound C ?? Hx l);
|1: apply (le_transitive ???? (H2 O));
apply (segment_upperbound ? l u);]
|2: intros;
- lapply (downarrow_lowerlocated ? a â\8c©x,hâ\8cª) as Ha1;
+ lapply (downarrow_lowerlocated ? a â\89ªx,hâ\89«) as Ha1;
[2: apply segment_preserves_downarrow;split; assumption;]
- lapply (segment_preserves_infimum ?l u a â\8c©?,hâ\8cª) as Ha2;
+ lapply (segment_preserves_infimum ?l u a â\89ª?,hâ\89«) as Ha2;
[2:split; assumption]; cases Ha2; clear Ha2;
cases (H1 a a); lapply (H7 H4 Ha1) as HaC;
lapply (segment_cauchy ? l u ? HaC) as Ha;