Let (X 1 , d 1 ) and (X 2 , d 2 ) be metric spaces and suppose f : X 1 → X 2 . Mark each statement True or False. Justify each answer. (a) To show that a sequence (s n ) converges to s in (X 1 , d 1...

Let (X₁, d₁) and (X₂, d₂) be metric spaces and suppose f : X₁
→ X₂. Mark each statement True or False. Justify each answer.

(a) To show that a sequence (s_n) converges to s in (X₁, d₁), it suffices to find a positive real sequence (a_n) such that d₁
(s_n, s) ≤ a_n
for all n and an → 0.

(b) The distance function d is continuous on X₁.

(c) If f is continuous at c ∈ X₁, then x n → c in X₁
whenever f (x n) → f (c) in X₂.