 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2023/prelims/mathematics-ii/5/a/i/solution

(f_{n})

f

E

\forall ε > 0 \exists N \forall n \geq N \forall x \in E : ∣ f_{n} (x) - f (x) ∣ < ε .

The same

N

must work for every

x

. Equivalently, the supremum norm of

f_{n} - f

tends to zero. This is stronger than pointwise convergence, where

N

may depend on

x

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