equal
deleted
inserted
replaced
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}"} |