Let (𝑋,𝑑𝑋) and (𝑌,𝑑𝑌) be a metric spaces with 𝑋 compact, with 𝑓 :𝑋 →𝑌 a continuous mapping between them. Let 𝜖 >0. By continuity, for every 𝑥 ∈𝑋 there exists 𝛿𝑥(𝜖) >0 such that 𝑓(𝑦) ∈B𝜖/2(𝑓(𝑥)) for all 𝑦 ∈B𝛿𝑥(𝜖)/2(𝑥). Then the balls {B𝛿𝑥(𝜖)/2(𝑥) :𝑥 ∈𝑋} form a open cover of 𝑋, so by compactness there must exist a finite set of points (𝑥𝑖)𝑛𝑖=1 whose balls cover the space, i.e. {B𝛿𝑥𝑖(𝜖)/2(𝑥𝑖)}𝑛𝑖=1 is a finite subcover. Then there exists 𝛿(𝜖) =min{12𝛿𝑥𝑖(𝜖)}𝑛𝑖=1 since it is the minimum of finitely many positive real numbers.

Now we will show that 𝛿(𝜖) meets the requirements for Uniform continuity. Let 𝑥,𝑦 ∈𝑋 such that 𝑑𝑋(𝑥,𝑦) <𝛿(𝜖). Then 𝑥 ∈B𝛿𝑥𝑖(𝜖)/2(𝑥𝑖) for some 0 ≤𝑖 ≤𝑛. Then by the triangle inequality

Therefore 𝑥,𝑦 ∈B𝛿𝑥𝑖(𝜖)(𝑥𝑖) and thus by the original definition of 𝛿𝑥𝑖(𝜖) it follows 𝑓(𝑥),𝑓(𝑦) ∈B𝜖/2(𝑓(𝑥𝑖)). Thus 𝑑(𝑓(𝑥),𝑓(𝑦)) <𝜖 as required.