Let(X, d) be a metric space. the set {y ∈ X : d(x, y) ≤ r} is a closed ball centered at X and with radius r.
(a)Show that a closed ball is a closed set.
Denote the closed ball centered at of radius as . Now suppose that was a limit point of but not an element of . Then every open ball around would contain another point of besides . Therefore . Assume that , then choosing would derive a contradiction. Therefore and the conclusion follows.