 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-4/23h/a/i/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

x = 0

, the claim is immediate. Otherwise weak convergence gives

∥ x ∥^{2} = ⟨ x, x ⟩ = lim_{n \to \infty} ⟨ x_{n}, x ⟩ .

By Cauchy-Schwarz,

⟨ x_{n}, x ⟩ \leq ∥ x_{n} ∥ ∥ x ∥ .

Taking the lower limit and dividing by

∥ x ∥

∥ x ∥ \leq n lim inf ∥ x_{n} ∥ .

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