Let M be some nonempty set

a) Let d be a metric on M and P ∈ M; Q ∈ B (P, r) . Show that

B (Q, s) ⊂ B (P, r + s)

b) Let d be a metric on M and P,Q ∈ M, T ∈ B (P, r) ∩ B (Q, r) . Find

some s > 0 so that B (T, s) ⊂ B (P, r) ∩ B (Q, r)

c) Suppose that d₁ and d₂ are two metrics on the set M such that for

all x, y ∈ M d₁ (x, y) ≤ 2d₂ (x, y) . Denote by B₁ (p, r) and B₂ (p, r)

the open balls with center p and radius r with in the metrics d₁ and

d₂ respectively. Show that B₂ (p, r) ⊂ B₁ (p, 2r)

If you help me, I will be really happy.