Uniform Cauchy and convergence clarification.

Question

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.

Answer 1

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.