This shows the graph of a certain sequence of functions over the interval [0,1]; it shows the graph of f_n(x)= x e^-nx for the indicated value of n.

What to do:

Change the value of x by moving the vertical grey rectangle to the left or right (by clicking and dragging with the mouse).

Increase n or reset n to 1 by clicking on the buttons on the lower left.

Notice that as you increase n, the value of f_n(x) for the current value of x eventually goes to 0. But also notice that the graph of f_n gets close to zero over the entire interval [0,1], as n increases (move x back and forth to see this). So this sequence of functions converges uniformly to 0 on [0,1].