Originally Posted by

**kingwinner** Suppose that (x_n) is a sequence in a metric space X such that lim x_n = a exists. Prove that {x_n: n E N} U {a} is a closed subset of X.

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Let B(r,a)={x E X: d(x,a) < r} denote the open ball of radius r about a.

__Definition:__ Let D be a subset of X. By definition, D is **open** iff for all a E D, there exists r>0 such that B(r,a) is contained in D.

__Definition:__ Let F be a subset of X. F is called **closed** iff whenever (x_n) is a sequence in F which converges to a E X, then a E F. (i.e. F contains all limit points of sequences in F)

__Theorem:__ F is closed in X iff the complement of F is open.

I know the definitions, but I just don't know out how to construct the proofs formally. I am not sure how to prove that some set is "closed"...I can't find any similar examples in my textbook.

May someone kindly help me out?

Any help is appreciated!

[note: also under discussion in math links forum]