src/HOL/Real_Vector_Spaces.thy
changeset 63680 6e1e8b5abbfa
parent 63556 36e9732988ce
child 63915 bab633745c7f
equal deleted inserted replaced
63679:dc311d55ad8f 63680:6e1e8b5abbfa
  2082 
  2082 
  2083 subsection \<open>The set of real numbers is a complete metric space\<close>
  2083 subsection \<open>The set of real numbers is a complete metric space\<close>
  2084 
  2084 
  2085 text \<open>
  2085 text \<open>
  2086   Proof that Cauchy sequences converge based on the one from
  2086   Proof that Cauchy sequences converge based on the one from
  2087   @{url "http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html"}
  2087   \<^url>\<open>http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html\<close>
  2088 \<close>
  2088 \<close>
  2089 
  2089 
  2090 text \<open>
  2090 text \<open>
  2091   If sequence @{term "X"} is Cauchy, then its limit is the lub of
  2091   If sequence @{term "X"} is Cauchy, then its limit is the lub of
  2092   @{term "{r::real. \<exists>N. \<forall>n\<ge>N. r < X n}"}
  2092   @{term "{r::real. \<exists>N. \<forall>n\<ge>N. r < X n}"}