# Uniform Cauchy and convergence clarification.

I have a doubt in the definition of uniform Cauchy sequence of functions in the domain $$A$$.
We say that the sequence $$(f_n)_{n\in\mathbb{N}}$$ is uniformly Cauchy when for any positive $$\epsilon$$ we have a $$n_1\in\mathbb{N}$$ such that $$\forall n,m\in\mathbb{N}$$ with $$n,m>n_1$$, $$\forall x\in A$$ it has $$|f_n(x)-f_m(x)|<\epsilon$$.

My doubt is whether it is possible to use to different values from the domain in this definition. That is "... $$\forall x,y\in A, |f_n(x)-f_m(y)|<\epsilon$$ "..

Furthermore I have the same problem for uniform convergence too.

Consider $$f_n(x)=x$$, $$|f_n(0)-f_m(1)|=1$$ but the sequence of functions converges uniformly since it is constant $$f_n(x)$$ so in the definition of uniformly convergence, $$x$$ is fixed.