# Cauchy Sequence and Completeness

tylerc1991

## Homework Statement

let (X,d) be a metric space and let A be a dense subset of X such that every Cauchy sequence in A converges in X. Prove that (X,d) is complete.

## Homework Equations

(X,d) is complete if all Cauchy sequences in X converge.
A is a dense subset of X => closure(A) = X

## The Attempt at a Solution

Since for all x in A there exists <x_n> in A s.t. <x_n> -> x, x is an element of closure(A) so x is in X, but since all Cauchy sequences in A converge in X for all x in X, doesn't this mean that all Cauchy sequences in X converge in X? Thank you for your help anyone!

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