The aim here is to show some videos illustrating the difference between pointwise and uniformly convergent sequences of functions.

Example 0: at the point just gives the sequence . This viewpoint (ie, sequences arising from fixed x values) is shown in the left picture below. However we normally consider the sequence of functions as increases, and this is shown on the figure below right.

Example 1: in blue converges pointwise (and in fact uniformly) to in orange below.

You see in the above picture that the function values are tending to zero in the same manner across the x-axis. We can ignore the oscillations since . However now consider the following example, which shows that not all convergences are so uniform.

Example 2: The graph below shows for .

The animation stops at so you can see how slow the rate of convergence is near ! Nonetheless for each point the function is still converging and the limit function is for and for . Indeed the limiting function is a rather unpleasant discontinuous one.

Uniform convergence arises because we would like to know whether there are points (like around above) where the function is converging really slowly and not at the same (i.e. uniform) rate as at other points.

Example 3: Now let’s consider the function on the range , which appears to be tending to the function .

However the red point above shows the maximum of this function at . This means the largest value of the expression and this is not a small number! In this example the point around which we have slow convergence changes with increasing . This is unlike Example 2 where the point was fixed for all (ie, ).

At the bottom here I’m pasting the generic Mathematica code used for the above:

Animate[Plot[(n*x)/(1 + (n*x)^2), {x, 0, 4}, Prolog -> {Red, PointSize[0.02], Point[{1/n, 1/2}]}, PlotRange -> {{0, 4}, {-0.2, 0.8}}], {n, 1, 15, 1, Appearance -> “Labeled”}];