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?

Intuitively, uniform convergence is like having a tubular neighbourhood of the limit function and the fuctions $f_n$ are definitely in the tube.
With pointwise convergence we simply ask that the succession $a_n:=f_n(x)$, $x$ fixed, tends to $f(x)$.
