Pointwise and uniform convergence definitions In my lecture notes it gives the following definition for pointwise and uniform convergence:
The sequence $f_k$ converges pointwise to a function $f$ on $A$ as $k \rightarrow\infty$  if the sequence $(f_k(x))_k\epsilon_\mathbb{N})$ converges to $f(x)$ for every $x\in A$; namely if for every $x \in A$ and every $\epsilon >0$ there exists $N\in \mathbb{N}$ such that for every $k \geq N$ we have $|f_k(x)-f(x)|<\epsilon$.
The sequence $f_k$ converges uniformly to $f$ on $A$ if 
$\forall \epsilon>0$ $\exists N \in \mathbb{N}$ such that
$\forall k \geq N, \forall x\in A: |f_k(x)-f(x)|<\epsilon$
I'm confused about the difference between the definition of pointwise and uniform convergence. They both seem the same to me. Could anyone clarify the difference please?

The key here is your $k.$ In pointwise convergence your $k \equiv k(x,\epsilon)$ but for uniform convergence it depends only on $\epsilon.$
