Let A be a bounded set in Rn Create a proof that the closure

Let A be a bounded set in R^n. Create a proof that the closure of A is compact in R^n.

Solution

Let X =Closure of A.

If A is finite , there is nothing to prove , as X =A. (every finite set if closed).

So let A be infinite.

Let (x_n) be an infinite sequence of points in X.

By Bolzano -Weierstrass theorem, every bounded sequence has a convergent subsequence.

Thus (x_n) has a convergent subsequence , converging to , say x

As X is closed ,, x belongs to X.

Thus X is sequentially compact. in R^n.

As Rⁿ is a metric space, it follows that X is compact in Rⁿ