closed ball always is closed in the meteric space

Q: Show that a closed ball is always closed in the meteric space?

This is my answer. Is it true?

Let t be any point in this complement of B(a;r). Compute the distance between the center of the closed ball and t; call it d.

By setting epsilon = d/2, the open neighborhood (ball) centered at t with radius epsilon is disjoint from the closed ball B(a;r).

Hence, the complement of the closed ball B(a;r) is open, thus the closed ball is a closed set.

could any one help me in this.

Re: closed ball always is closed in the meteric space

The basic idea is correct, except that is insufficient for the proof. For example, let and . Then . If you choose then you may get something like , in which case , which is not disjoint from .

Instead let , and this will suffice for the proof.