Definition BQpair_aux2b C := forall n m, natp n -> natp m -> P (Vg C n) <q Q (Vg C m) ,
every Cauchy sequence has a limit Consider first a Cauchy sequence of rational numbers y. Let B be the set of all t ? Q such that, for some N , y n < t when n ? N . If B has a least element b, then b is the limit of the sequence y. Otherwise B is a real number (note that B is non-empty since the sequence y is bounded), and is the limit of y. Let x be a real Cauchy sequence. We can find (via the axiom of choice) a sequence of rational numbers y such that |x n ? y n This sequence is then Cauchy ,
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