equal
deleted
inserted
replaced
86 then have nonempty: "{lim (?f n) |n. x < r n} \<noteq> {}" for x |
86 then have nonempty: "{lim (?f n) |n. x < r n} \<noteq> {}" for x |
87 by auto |
87 by auto |
88 |
88 |
89 define F where "F x = Inf {lim (?f n) |n. x < r n}" for x |
89 define F where "F x = Inf {lim (?f n) |n. x < r n}" for x |
90 have F_eq: "ereal (F x) = (INF n\<in>{n. x < r n}. ereal (lim (?f n)))" for x |
90 have F_eq: "ereal (F x) = (INF n\<in>{n. x < r n}. ereal (lim (?f n)))" for x |
91 unfolding F_def by (subst ereal_Inf'[OF bdd_below nonempty]) (simp add: setcompr_eq_image) |
91 unfolding F_def by (subst ereal_Inf'[OF bdd_below nonempty]) (simp add: setcompr_eq_image image_comp) |
92 have mono_F: "mono F" |
92 have mono_F: "mono F" |
93 using nonempty by (auto intro!: cInf_superset_mono simp: F_def bdd_below mono_def) |
93 using nonempty by (auto intro!: cInf_superset_mono simp: F_def bdd_below mono_def) |
94 moreover have "\<And>x. continuous (at_right x) F" |
94 moreover have "\<And>x. continuous (at_right x) F" |
95 unfolding continuous_within order_tendsto_iff eventually_at_right[OF less_add_one] |
95 unfolding continuous_within order_tendsto_iff eventually_at_right[OF less_add_one] |
96 proof safe |
96 proof safe |