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## Homework Statement

Let X be an incomplete metric space. and Let X' denote its completion. I would like to show that there is Cauchy sequence in X which does not converge in X but does converge in X'. Moreover, I want to show that X contains every element of the sequence except the limit point.

## Homework Equations

No equations as such other than the

Definition: A complete metric space is one in which every Cauchy sequence converges.

Definition: A Cauchy sequence { x_n } is one for which given any epsilon > 0 there exists a natural N such that if m,n > N then d(x_n,x_m) < epsilon.

## The Attempt at a Solution

Well, negating the definition of a metric space we should be able to find a cauchy sequence which does not converge in X.

My next line of argument would be to say that since X is isometrically embedded in X'. Then { x_n } is also a Cauchy sequence in X' and thus converges in X'.

However, I'm not sure if it is always the case that X' \ X will always only contain the limit point. Can it not contain other points in the sequence?

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