Let (X, ||⋅||) be a normed vector space. Prove the following properties:
(a) The induced distance d(x₁, x₂) = ||x₁ - x₂|| is a metric.
(b) | ||x₂|| - ||x₁|| | ≤ ||x₂ - x₁|| (Hint: try treating ||x₁|| ≤ ||x₂|| cases separately)
(c) The norm ||⋅||: X → ℝ is a continuous function from the metric space (X, d), where d(x₁, x₂) = ||x₁ - x₂|| is the induced metric, to the real numbers.