I'm looking at http://en.wikipedia.org/wiki/It%C5%8D%27s_formula#Informal_derivation where it says dB^2 tends to E(dB^2). I followed the link to the basic properties for Wiener processes, but I can't find why dB^2 tends to E(dB^2). I am guessing that it has to do with the limit as dt approaches zero. The closest thing seems to be that the variance of a Wiener process is t, but that's not quite the same thing. dB is a sampling of a normal random variable, it is not a summary statistic.
For context, I am looking at the Ito Lemma for Geometric Brownian motion (immediately above the Ito derivation link above). In the second line, there is a -(1/2)(sigma^2)dt. This is a direct result of the fact that the random variable dB^2 gets replaced by dt. It seems to be a pivotal change, so I'd like to understand it.