This shows the graph of a certain sequence of functions over the interval [0,1]; it shows the graph of fn(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 fn(x) for the current value of x eventually goes to 0, but the graph of fn 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.