equal
deleted
inserted
replaced
2789 qed |
2789 qed |
2790 |
2790 |
2791 lemma closed_contains_Inf: |
2791 lemma closed_contains_Inf: |
2792 fixes S :: "real set" |
2792 fixes S :: "real set" |
2793 shows "S \<noteq> {} \<Longrightarrow> bdd_below S \<Longrightarrow> closed S \<Longrightarrow> Inf S \<in> S" |
2793 shows "S \<noteq> {} \<Longrightarrow> bdd_below S \<Longrightarrow> closed S \<Longrightarrow> Inf S \<in> S" |
2794 by (metis closure_contains_Inf closure_closed assms) |
2794 by (metis closure_contains_Inf closure_closed) |
2795 |
2795 |
2796 lemma closed_subset_contains_Inf: |
2796 lemma closed_subset_contains_Inf: |
2797 fixes A C :: "real set" |
2797 fixes A C :: "real set" |
2798 shows "closed C \<Longrightarrow> A \<subseteq> C \<Longrightarrow> A \<noteq> {} \<Longrightarrow> bdd_below A \<Longrightarrow> Inf A \<in> C" |
2798 shows "closed C \<Longrightarrow> A \<subseteq> C \<Longrightarrow> A \<noteq> {} \<Longrightarrow> bdd_below A \<Longrightarrow> Inf A \<in> C" |
2799 by (metis closure_contains_Inf closure_minimal subset_eq) |
2799 by (metis closure_contains_Inf closure_minimal subset_eq) |