1. Let (X, d) be a metric space. For all x, y ∈ X define d 4 (x,y) = Verify that d 4 is a metric on X. [Note that d 4 is a bounded metric on X in the sense that 0 ≤ d 4 (x, y) ≤ 1 for all x, y ∈ X.]...

1. Let (X, d) be a metric space. For all x, y ∈ X define

d₄(x,y) =

Verify that d₄
is a metric on X. [Note that d₄
is a bounded metric on X in the sense that 0 ≤ d₄(x, y) ≤ 1 for all x, y ∈ X.]

2. If A and B are compact subsets of a metric space (X, d ), prove that A ∪ B is compact.