I know that GBM can be discretely approximated by methods such as Euler-Maruyama, and it can be shown that Binomial tree converges to GBM at the continuous time limit.
However I'm having a hard time to understand the intuition. If a GBM process behaves like a binomial tree, then wouldn't the log diffrences have only two possible outcomes?
My understanding is that time step is the key. Suppose the infetestimal step is hourly, but when we observe daily data points, there are already $2^{24}$ possible outcomes and these daily observations are log-normal. Comments?