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\bDefinition 1. Cauchy relation \^ch. b\ For two sequences a\[n \ and b\[n \ ,
              a\^chb : \^.fa\^ep  \^.exn  \^.fan',n'', n'&gt;n, n''&gt;n |a\[n' \ - b\[n'' \ | &lt; \^ep.


\bNotation.b\  \^.exm P(m',m'') 
           will denote
          \^.exm  \^.fam',m'', m'&gt;m, m''&gt;m P(m',m'') 
           like in above definition.
           If context is clear, "\^.exm"  can be also omitted.
\bDefinition 2. a is Chauchy sequence iff b\ a\^cha.

There are equivalent definitions and properties of Cauchy sequences:
   1. a\^chb =&gt; a and b are Cauchy sequences.
   
\bDefinition 1'.b\ a\^chb : sequence a\[1 \  b\[1 \  a\[2 \  b\[2 \  .... is Cauchy.

   2. An arbitrary change of order in sequences a and b or taking subsequences of a and b
      does not change Cauchy relation between a and b.



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