can someone explain the difference in uniform convergence and pointwise convergence

Question

In wiki I found this explanation:

for uniform convergence: $\forall \varepsilon >0\ \underline {\exists N\in \mathbb{N} \ \forall x\in D_{f}}\ \forall n\geq N:\quad \left|f_{n}(x)-f(x)\right|<\varepsilon ,$

for pointwise convergence: $\forall \varepsilon >0\ \underline {\forall x\in D_{f}\ \exists N\in \mathbb{N} }\ \forall n\geq N:\quad \left|f_{n}(x)-f(x)\right|<\varepsilon ,$

I'm trying to understand what the difference in the order of the two underlined parts exactly mean. And why that implies that pointwise convergence follows from uniform convergence.

Answer 1

They already told you the formal things. The point is you cannot have a good understanding of this concept without visualising it graphically.

For this purpose, I'll give you two sequences of functions $(f_n) _n$ and $(g_n) _n$. The first one will converge pointwise, the second will converge uniformly (and therefore also pointwise).

You should DRAW two graphs, one for the $f_n$ and one for the $g_n$. Ready?

• $f_n: (0,1) \rightarrow \mathbb{R}$ such that $f_n(x)=x^n$

• $g_n: (0,1) \rightarrow \mathbb{R}$ such that $g_n(x)=\frac{x}{n} $

for $n\ge 0$. Note that you should only consider the domain $(0,1)$ for our purpose.

Now draw the graphs for the first $n$'s. You see both sequences converge to the zero function $f: (0,1) \rightarrow \mathbb{R}$ such that $f(x)=0$

The point is, since $f_n(1)=1$ for all $n$, at every fixed $n$ the maximum distance between $f$ and $f_n$ remains constantly 1, thus the sequence does not UNIFORMLY converge.

On the other hand, the $g_n$'s get pushed towards zero everywhere (it does not happen anymore that one extreme remains fixed at the same height). Thus, they converge uniformly.

The limit function $f$ being flat in this simple case, you can visualize the uniform convergence condition as "as $n$ grows, I can squeeze the graphs of $g_n$'s to zero with a horizontal line", which I can't do with the other sequence. The height of the horizontal line is the $\epsilon$ of your (first) definition.

Once this concept becomes clear, if you try and write down the formal definitions, you end up with the two (ugly) lines that you wrote.

EDIT: I'm happy I've been chosen as the main answer, thanks! If anyone finds it useful please leave an upvote :)

Answer 2

In the definition of pointwise convergence, the $N$ is chosen after $x$ is chosen, so it may depend on $x$ (there is no guarantee that the same $N$ works for every $x$). The definition of uniform convergence is stronger, because it states that there is a single $N$ that works for all $x$.

Edit. Uniform convergence implies pointwise convergence because, of course, even knowing $x$ we can still choose the same $N$ that the definition of uniform convergence gave us. (We don't need to have it depend on $x$.)