 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ib/paper-1/10g/d/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

This is always true. Uniform convergence makes

(f_{n})

ε > 0

, there is

N

such that

d_{Y} (f_{n} (x), f_{m} (x)) < ε

for every

x \in A

and

m, n \geq N

. Letting

x \to x_{0}

through points of

A ∖ {x_{0}}

and using the assumed limits gives

d_{Y} (y_{n}, y_{m}) \leq ε .

Thus $(y_{n})$ is a Cauchy sequence. Since $Y$ is a complete metric space, $(y_{n})$ converges in $Y$ .

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