 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ib/paper-2/12e/a/iii/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

Suppose

c ∥ v ∥ \leq ∥ v ∥^{'} \leq C ∥ v ∥.

(v_{n})

∥ \cdot ∥

, then

∥ v_{n} - v_{m} ∥^{'} \leq C ∥ v_{n} - v_{m} ∥ ⟶ 0.

Conversely,

∥ v_{n} - v_{m} ∥ \leq c^{- 1} ∥ v_{n} - v_{m} ∥^{'} \to 0

. Hence the sequence is Cauchy for one norm exactly when it is Cauchy for the other.

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