un(x)u_n(x)un(x) in x∈Ax\in Ax∈A is said to be uniformly Cauchy iff: ∀ϵ>0∃N∈Z+∀m,n>N∀x∈A;∣un(x)−um(x)∣<ϵ\forall \epsilon \gt 0 \exists N \in \mathbb{Z}^+ \forall m,n \gt N \forall x \in A; \lvert u_n(x)-u_m(x) \rvert \lt \epsilon ∀ϵ>0∃N∈Z+∀m,n>N∀x∈A;∣un(x)−um(x)∣<ϵ If un(x)u_n(x)un(x) is a sequence of real-valued functions, then: un(x) converges uniformly ⟺ un(x) is uniformly Cauchyu_n(x)\text{ converges uniformly} \iff u_n(x)\text{ is uniformly Cauchy} un(x) converges uniformly⟺un(x) is uniformly Cauchy