Understanding how sequences of functions change and behave is important in calculus. Let's talk about two types of convergence: pointwise and uniform convergence.

Pointwise Convergence:

A sequence of functions, which we'll call $\{f_n(x)\}$, converges pointwise to another function $f(x)$ on a set of numbers called a domain $D$ if, for every point $x$ in $D$, the values of $f_n(x)$ get closer to $f(x)$ as we keep adding more functions, or as $n$ becomes larger.

This means that if you pick any $x$ in our domain, and look at the function $f_n$ at that point, it will eventually get really close to $f(x)$.

Example of Pointwise Convergence:

Let's look at a simple example. Suppose our functions are defined as $f_n(x) = \frac{x}{n}$.

As $n$ gets bigger, for every specific value of $x$, $f_n(x)$ will get closer to 0. However, the speed at which it gets there can be different depending on the size of $x$. For larger values of $x$, $f_n(x)$ can stay bigger for a longer time before it finally goes down to 0.

You can imagine drawing this on a graph. For a fixed value of $x$, you can draw a line to show how $f_n(x)$ moves down toward the value of $f(x)$. But remember, pointwise convergence means that each point can behave a bit differently.

Uniform Convergence:

Now, let's talk about uniform convergence. This is a bit different. When we say that a sequence converges uniformly to $f(x)$, we mean that the distance between $f_n(x)$ and $f(x)$ gets small in the same way for all $x$ at the same time.

Example of Uniform Convergence:

A good example to understand this is $f_n(x) = x^n$ on the range [0, 1].

As $n$ increases, $f_n(x)$ pointwise goes to 0 for $x$ values between 0 and 1. But at $x = 1$, it stays at 1.

Here, we notice that while the functions are coming closer to $f(x)$, the maximum distance from $f_n(x)$ to $f(x)$ doesn't shrink uniformly as $x$ changes.

To see this on a graph of uniform convergence, picture lines that show how far $f_n(x)$ is from $f(x)$, and you’d notice that the gaps between them shrink evenly for every $x$ as $n increases.

In summary, pointwise convergence allows points to move towards a limit at different speeds, while uniform convergence means everything is closing in together evenly.