<Li> Every Cauchy sequence of real (or complex) numbers is bounded (since for some N, all terms of the sequence from the N - th onwards are within distance 1 of each other, and if M is the largest absolute value of the terms up to and including the N - th, then no term of the sequence has absolute value greater than M+1). </Li> <Li> In any metric space, a Cauchy sequence which has a convergent subsequence with limit s is itself convergent (with the same limit), since, given any real number r> 0, beyond some fixed point in the original sequence, every term of the subsequence is within distance r / 2 of s, and any two terms of the original sequence are within distance r / 2 of each other, so every term of the original sequence is within distance r of s . </Li> <P> These last two properties, together with the Bolzano--Weierstrass theorem, yield one standard proof of the completeness of the real numbers, closely related to both the Bolzano--Weierstrass theorem and the Heine--Borel theorem . Every Cauchy sequence of real numbers is bounded, hence by Bolzano - Weierstrass has a convergent subsequence, hence is itself convergent . It should be noted, though, that this proof of the completeness of the real numbers implicitly makes use of the least upper bound axiom . The alternative approach, mentioned above, of constructing the real numbers as the completion of the rational numbers, makes the completeness of the real numbers tautological . </P> <P> One of the standard illustrations of the advantage of being able to work with Cauchy sequences and make use of completeness is provided by consideration of the summation of an infinite series of real numbers (or, more generally, of elements of any complete normed linear space, or Banach space). Such a series ∑ n = 1 ∞ x n (\ displaystyle \ sum _ (n = 1) ^ (\ infty) x_ (n)) is considered to be convergent if and only if the sequence of partial sums (s m) (\ displaystyle (s_ (m))) is convergent, where s m = ∑ n = 1 m x n (\ displaystyle s_ (m) = \ sum _ (n = 1) ^ (m) x_ (n)). It is a routine matter to determine whether the sequence of partial sums is Cauchy or not, since for positive integers p> q, </P>

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