Am I right to write the definition of uniform convergence as follows?
$(f_n)$ converges to $f$ uniformly if and only if $$\forall\epsilon>0, \exists N\in\mathbb{N}, \forall n, \forall x (n\geq N \Rightarrow |f_n(x)-f(x)|<\epsilon) $$
I am not sure where should I put $\forall x$.
The uniform convergence means that starting from a certain $N$, all $f_n$ functions are included in a "band"(yellow one bellow), they are approaching the limit function $f$ uniformly, i.e, the functions $f_n$ are approaching the limit globally. Therefore, starting from $N$ all the values of the function $f_n$ for a certain $n$ are in the band. That is what you said : $ \forall n\geq N, \forall x : |f_n(x)-f(x)|<\epsilon$ .