Convergence, Cauchy sequences and completeness

2 Convergence, Cauchy sequences and completeness We begin with several definitions. 

Definition 31 In a metric space an infinite sequence of elements fi, f2, fs) said 

Definition 32 Any sequence in which the distance between any two elements tends to zero, — 0, as k,j — oo, is called a Cauchy sequence. 

It can be proved that any convergent sequence is a Cauchy sequence. In fact, from the triangle inequality, we can write 

On the other hand, there can exist Cauchy sequences of elements which do not converge to an element in the metric space. For example, let us consider as a metric space the internal part of the geometric 3-D ball B without a boundary. We can introduce series of points Si, 2, S3. which converge to the element Sq located at the boundary. Obviously, the set 81,82,83. forms a Cauchy sequence, but it converges to the element sq outside our metric space B. From this point of view we can call B an incomplete metric space. 

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