classical mechanics - Derivation of the Lagrangian method using discretized time axis

I'm watching this video lecture by Leonard Susskind of Stanford: http://www.youtube.com/watch?v=3apIZCpmdls

After some preliminaries, at 34 minutes he jumps into a discretization of the time axis and proceeds to mess around with it and the Lagrangian, completely losing me in the process. I haven't come across this discrete method before. What is this called and are there any resources on the net (or textbooks) that use this approach rather than a continuous approach? Is it necessary to do this discretely rather than continuously?