src/HOL/Library/Extended_Real.thy
changeset 63145 703edebd1d92
parent 63099 af0e964aad7b
child 63225 19d2be0e5e9f
equal deleted inserted replaced
63143:ef72b104fa32 63145:703edebd1d92
  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