equal
deleted
inserted
replaced
3550 shows "(f \<longlongrightarrow> -\<infinity>) net \<longleftrightarrow> Limsup net f = -\<infinity>" |
3550 shows "(f \<longlongrightarrow> -\<infinity>) net \<longleftrightarrow> Limsup net f = -\<infinity>" |
3551 unfolding tendsto_iff_Liminf_eq_Limsup[OF assms] |
3551 unfolding tendsto_iff_Liminf_eq_Limsup[OF assms] |
3552 using Liminf_le_Limsup[OF assms, of f] |
3552 using Liminf_le_Limsup[OF assms, of f] |
3553 by auto |
3553 by auto |
3554 |
3554 |
3555 lemma convergent_ereal: -- \<open>RENAME\<close> |
3555 lemma convergent_ereal: \<comment> \<open>RENAME\<close> |
3556 fixes X :: "nat \<Rightarrow> 'a :: {complete_linorder,linorder_topology}" |
3556 fixes X :: "nat \<Rightarrow> 'a :: {complete_linorder,linorder_topology}" |
3557 shows "convergent X \<longleftrightarrow> limsup X = liminf X" |
3557 shows "convergent X \<longleftrightarrow> limsup X = liminf X" |
3558 using tendsto_iff_Liminf_eq_Limsup[of sequentially] |
3558 using tendsto_iff_Liminf_eq_Limsup[of sequentially] |
3559 by (auto simp: convergent_def) |
3559 by (auto simp: convergent_def) |
3560 |
3560 |