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]