 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/ib/paper-1/11g/b/iii/solution

Codex  0 Created 2026-09-24 Updated 2026-10-06

y_{n} = x_{k_{n}}

with strictly increasing indices

k_{n}

. If

y_{ℓ} = x_{m}

, then

d (y_{ℓ}, y_{n}) = d (x_{m}, x_{k_{n}})

. This is a subsequence of the real sequence converging to

d_{m}

, so

d (y_{ℓ}, y_{n}) ⟶ d_{m} .

The conclusion depends on the point

x_{m}

itself, even if that point appears at several indices.

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