Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics II / 5 / a / i

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- Solution i

Solution

The sequence of functions

(f_{n})

converges uniformly to

f

on

E

if

\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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