Prove: Interior is an open set

Let (X,d) be a metric space and let F be a subset of X. B(r,x) is the open ball of radius r about x.

__Definition:__ The **interior** of F, int(F), is the set of all x E F such that there is an r > 0 such that B(r,x) is contained in F.

__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.

**Using these definitions, prove that int(F) is an open set.**

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Attempt:

Let x E int(F), then there exists r>0 such that B(r,x) is contained in F.

We need to prove that there exists some r' >0 s.t. B(r',x) is contained in int(F). How can we prove this?

Any help is greatly appreciated!