+lemma restrict_uniform_convergence_uparrow:
+ ∀C:ordered_uniform_space.property_sigma C →
+ ∀l,u:C.exhaustive {[l,u]} →
+ ∀a:sequence {[l,u]}.∀x:C. ⌊n,\fst (a n)⌋ ↑ x →
+ x∈[l,u] ∧ ∀h:x ∈ [l,u].a uniform_converges ≪x,h≫.
+intros; cases H2 (Ha Hx); clear H2; cases Hx; split;
+[1: split;
+ [1: apply (supremum_is_upper_bound C ?? Hx u);
+ apply (segment_upperbound ? l);
+ |2: apply (le_transitive ? ??? ? (H2 O));
+ apply (segment_lowerbound ?l u);]
+|2: intros;
+ lapply (uparrow_upperlocated ? a ≪x,h≫) as Ha1;
+ [2: apply segment_preserves_uparrow;split; assumption;]
+ lapply (segment_preserves_supremum ? l u a ≪?,h≫) 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;
+ lapply (sigma_cauchy ? H ? x ? Ha); [left; split; assumption]
+ apply restric_uniform_convergence; assumption;]
+qed.
+
+lemma restrict_uniform_convergence_downarrow:
+ ∀C:ordered_uniform_space.property_sigma C →
+ ∀l,u:C.exhaustive {[l,u]} →
+ ∀a:sequence {[l,u]}.∀x:C. ⌊n,\fst (a n)⌋ ↓ x →
+ x∈[l,u] ∧ ∀h:x ∈ [l,u].a uniform_converges ≪x,h≫.
+intros; cases H2 (Ha Hx); clear H2; cases Hx; split;
+[1: split;
+ [2: apply (infimum_is_lower_bound C ?? Hx l);
+ apply (segment_lowerbound ? l u);
+ |1: apply (le_transitive ???? (H2 O));
+ apply (segment_upperbound ? l u);]
+|2: intros;
+ lapply (downarrow_lowerlocated ? a ≪x,h≫) as Ha1;
+ [2: apply segment_preserves_downarrow;split; assumption;]
+ lapply (segment_preserves_infimum ?l u a ≪?,h≫) 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;
+ lapply (sigma_cauchy ? H ? x ? Ha); [right; split; assumption]
+ apply restric_uniform_convergence; assumption;]
+qed.
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