I am having a great deal of difficulty visualizing and grasping the concepts of box, product, and uniform topologies. Is there a clear, concise-ish way to explain it?
Also, I sort of understand the concepts of converging sequences under these topologies, but am lost. How does one determine whether or not a sequence converges with respect to different topologies?
Convergence in the product topology is equivalent to coordinatewise convergence. The above sequence converges coordinatewise to (0,0,0,0,...). But for it to converge uniformly, you would have to show that, after a certain point in the sequence, all the coordinates of each point were close to 0, and clearly that is not going to happen.