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## Homework Statement

Show that ##\overline{B(a,r)} = \{ x \in \mathbf R^n ; |a-x| \le r \}## in ##\mathbf R^n## for all points ##a \in \mathbf R^n## and ##r>0##.

Is it possible to generalize the statement to any normed vector space?

Give a example of a metric space where the statement is not true.

## Homework Equations

Definition:

If ##X## is a matric space, if ##E \subset X##, and if ##E'## denotes the set of all limit points of ##E## in ##X##, then the closure of ##E## is the set ##\bar E = E \cap E'##.

Definition:

Let ##(M,d)## be a metric space. An open ball with center in ##\mathbf a \in M## and radius ##r>0## is the set

##B(a,r) = \{x \in M;d(x,a)< r\}##.

## The Attempt at a Solution

Starting with the first part:

In ##\mathbf R^n## the open ball becomes

##B(a,r) = \{x \in M; ||\mathbf a -\mathbf x|| < r\}##.

Since ##\bar B## is union of all limits point and the ##B## I need to show that an arbitrary point

##p\in \{x \in M; ||\mathbf a -\mathbf x|| = r\}## is a limit point

while

##p\in \{x \in M; ||\mathbf a -\mathbf x|| > r\}## isn't one.

Take any ##p\in \{x \in M; ||\mathbf a -\mathbf x|| = r\}##. ##p## is a limit point of ##B## if every open ball to ##p## contains a point ##q \ne p## such that ##q \in B##.

That is if for all ##\epsilon > 0## there is a point ##q \in \{x\in M; ||p-x||<\epsilon \} \cap B## (since ##p## and ##q## are from two disjoint sets).

This seems to be to be true since "for any ##\epsilon > 0## there is an infinite number of real numbers in the open ball" but I'm not sure what is expected of a proof or how to prove it.