There is a choice of equivalent definitions of a closed set to use for the proof
(see for instance http://mathworld.wolfram.com/ClosedSet.html).
I suggest to use the following definition (4th in the page recommended) as the most convenient for the proof requested:
A set S is closed if every point outside S has a neighborhood disjoint from (outside of) S.
Now we look at the closed ball. Any point y outside of B[x;r] has
d(x,y) > r (1)
(by the definition of ...
The solution proves closure of a ball.