Any Cauchy sequence is bounded.
Exercise 1.99 showed that every convergent sequence is a Cauchy sequence; that is, the terms of the sequence become arbitrarily close to one another. The converse is not always true. There are metric spaces in which a Cauchy sequence does not converge to an element of the space. A complete metric space is one in which every Cauchy sequence is convergent. Roughly speaking, a metric space is complete if every sequence that tries to converge is successful, in the sense that it finds its limit in the space. It is a fundamental result of elementary analysis that the set Rn is complete; that is, every Cauchy sequence of real numbers converges. This implies that Rn is complete (exercise 1.211).