(A) (4 marks) Prove that for all x, y € Rn
d[infinity] (x, y) ≤d₂(x, y) ≤ √nd[infinity]. (x, y).
(B) (4 marks) Let xn, be a sequence of points in R² given by xn = 1/2^n , 1 - 1/3^n , . Prove that the sequence converges to the point (0, 1) in the metric d[infinity] on R². Does this sequence converge in the metric d₂?
(C) (4 marks) Let (X, d) be a metric space, and equip the cartesian product X x X with the product metric
dp((x1, y₁), (x2, y2)) = d(x1, x₂) + d(y₁, y2)
for all (x1, y₁) and (x2, y2) in X x X.
Let f: X X X → R be a map defined by
f(x, y) = d(x, y),
for all (x, y) € X x X. The space R of the reals is equipped with the standard metric. Prove that f is continuous with respect to the given metrics.
(D) (4 marks) Prove that the initial value problem
y' =- (3t² )/2 -(y - 1); y(0) = 0
has a unique solution in the space C[0, 1]. Here the space C[0, 1] is equipped with the d[infinity] metric.