TITLE: The difference between pointwise convergence and uniform convergence of functional sequences
QUESTION [46 upvotes]: $f_n$ converges pointwise to $f$ on $E$ if $\forall x \in E$ and $\forall \varepsilon > 0$, $\exists N \in \mathbb N$, such that $\forall n \geq N$ we have $\left|f_n(x) − f(x)\right| < \varepsilon$.
$f_n$ converges uniformly to $f$ on $E$ if  $\forall \varepsilon > 0$, $\exists N \in \mathbb{N}$ so that $\forall n \geq N$ we have $\left|f_n(x) − f(x)\right| < \varepsilon$ which holds for all $x \in E$.
I know the difference in definition, pointwise convergence tells us that for each point and each epsilon, we can find an $N$ (which depends from $x$ and $\varepsilon$)so that ... and the uniform convergence tells us that for each $\varepsilon$ we can find a number $N$ (which depends only from $\varepsilon$) s.t. ... .
But it seems again to me a bit fuzzy. Can somebody explain the difference more 'deeper' or more 'philosophically'? Or may be illustrate it visually?

REPLY [10 votes]: I understand why you find it fuzzy. I'll try to clarify it the way I see it:
I like to approach definitions by asking: Why were they thought of?
In this scenario, we have 2 very similar definitions that it's hard to visualize the difference between them.
I like to think of Uniform Convergence as the one that came first since it's the more intuitive of the two. In simple term, all it means is that $f_n(x)$ converges to $f(x)$ by only increasing $n$.
A simple example where it's easy to intuitively see the Uniform Converges is $f_n(x)=\sqrt{x^2+\frac{1}{n}}\to \sqrt{x^2}$. 
Why do we need Point wise convergence? Because sometimes this definitions breaks down and doesn't work:
For example: Let $D=(0,1)$ and $f_n(x)=x^n$. At first, it seems like $f_n(x)$ Uniformly Converges to $0$. But upon a closer inspection we notice that if $x(n)=\frac{1}{\sqrt[n]{3}}
\underset{n\to\infty}{\longrightarrow}1$ then $f_n(x)$ doesn't converge to $0$. In other words, we have to limit ourselves to $x$s such that an $N$ exists where for $n>N$ it converges (in other words, $N$ is also a function of $x$).
Notice how the Uniform Convergence broke down on the edge case of $x(n)\to 1$. This is the common theme for Pointwise convergence problems and is why (in my opinion) we have this definition.
To sum up: Uniform convergence is the intuitive definition. Pointwise Convergence was defined to handle the special cases where Uniform Convergence doesn't work.