Suppose that A0 A1 An are bounded subsets of a metric spac

Suppose that A_0, A_1, .. ., A_n are bounded subsets of a metric space (X, d). Show that their union A_0 U. .. U A_n is bounded.

Solution

Since each Ai is bounded, each Ai is contained in the ball of centre Oi and radius Ri.

Let l be the distance between O1 and O2.

Since y is in Y, d(y,O1) <=d(y,O2) +d(O2,O1) <=r2+l

<=r2+l+r1

Hence distance of each y from C1 is bounded and similarly distance of each x from C2 is bounded

In other words, A1UA2 is bounded.

Extending this to one more set, we get A1UA2 UA3 is bounded.

Repeating the process n+1 times we get A0 U.... An is bounded.