Do the following problems You should find the Sequence Lemma

Do the following problems. You should find the Sequence Lemma useful in the first x two.(Remember that the Sequence Lemma, in the context of metric spaces, says that *x epsilon Cl(A) if and only if there is a sequence a in A that converges to x.) Also, remember that there is a relationship between A and Cl(A) when A is closed. Let X be a metric space and let A X, where A is complete. Prove that A is closed in X. Let X be a complete metric space and let A be a closed subset of X. Prove that A is complete.

Solution

Let (x_n) be a Cauchy sequence of points in A . Then (x_n) also satisfies the Cauchy condition in X, and since X is complete, there exists x X such that x_n x. But A is also closed, so x A showing that A is complete.