This shows the graph of a certain sequence of functions over the interval [0,1]; it shows the graph of f_n(x)= nx 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 the graph of f_n is not close to zero over the entire interval [0,1]. (Move x back and forth to see this: the maximum of the function over [0,1] is e^-1 = 0.368 for all n.) So this sequence of functions converges pointwise to 0 but not uniformly.