# Pointwise vs. Uniform Convergence

This is a pretty basic question. I just don’t understand the definition of uniform convergence.

Here are my given definitions for pointwise and uniform convergence:

Pointwise convergence: Let $X$ be a set, and let $F$ be the real or complex numbers. Consider a sequence of functions $f_n$ where $f_n:X\to F$ is a bounded function for each $n\in \mathbb N$. $f:X\to F$ is the pointwise limit of $f_n$ if for every $x \in X$, $$\lim_{n\to \infty}f_n(x)=f(x).$$

Uniform convergence: Let $f_n$ be a sequence of functions in the set of all bounded functions from $X$ to $F$ where $F$ is the real or complex numbers. The sequence is said to converge uniformly to a bounded function $f:X \to F$ if, given $\epsilon>0$, there exists an $N\in \mathbb N$ s.t. $\sup\{|f_n(x)-f(x)| : x \in X \}<\epsilon$ for $n\ge N$

I’m sorry I don’t have a more specific question. I just don’t see the exact relation/difference between the two definitions. I’ve asked two different professors to explain this to me but neither of their explanations helped.

Edit: Attempting to show that uniform convergence implies pointwise convergence
if $f_n$ converges uniformly to f, then $\sup\{|f_n(x)-f(x)| : x\in X \}$ for $n\ge N$. Thus, $|f_n(x)-f(x)|<\epsilon$ for $n\ge N$, which is the definition of pointwise convergence.

Comparison

Pointwise convergence means at every point the sequence of functions has its own speed of convergence (that can be very fast at some points and very very very very slow at others).

Imagine how slow that sequence tends to zero at more and more outer points:
$$\frac{1}{n}x^2\to 0$$

Uniform convergence means there is an overall speed of convergence.

In the above example no matter which speed you consider there will be always a point far outside at which your sequence has slower speed of convergence, that is it doesn’t converge uniformly.

Another Approach

One can check uniform convergence by considering the “infimum of speeds over all points”. If it doesn’t vanish then it is uniformly convergent. And that gives another characterization as the ones with nonvanishing overall speed of convergence.