The concept of Brownian motion is usually preceded by first explaining a random walk, where someone starts at 0 on a number line, and at each point in time takes one step forward (with probability 1/2) or one step backward (probability 1/2). Then, they say something like, imagine instead of moving 1 step for every 1 unit increase in time, you move sqrt(h) steps in every h period of time, and take the limit as h goes to zero.

This, the very beginning, is where I begin to get confused. From t=0 to t=2 we would travel two steps. From t=2 to t=4 we would travel two steps. Yet, in h = 4 periods, we were supposed to travel sqrt(4) = 2 steps, not 4!

How is this resolved? Do things just get different from you move down into infinitesimal changes in distance and time?