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## Homework Statement

Prove that if lim n→∞ (p_n ) = p in a metric space then the set of points {p,p_1,p_2, ...,} are closed.

**2. Relevant information**

The definition of close in my book is "a set is closed if and only if its complementary is open." So I want to prove this by contradiction. I can't prove it by using accumulation points or compactness.

A theorem in my book states that a set S in a metric space is closed if and only if whenever {q_1,q_2 ,... } is a sequence of points in S that is convergent then the lim n→∞ q_n ∈S .

## The Attempt at a Solution

Suppose the set S is not closed which will imply that S^c is not open then there exists a point in S^c; lets denote this point p ∈ S^c such that there is an open ball B(p,r)∈ S^c ...

But I cant continue any further because I don't know what to do next.

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