I see the following statement made in a number of finance papers which indicates that the infinitesimal change in the Wiener process is proportional to the square root of (time), where the proportionality coefficient is normally distributed.

I cannot find a rigorous proof (or even a rudimentary proof) for this.

Am I missing something? Is it the case that dW is somehow just chosen to be $\displaystyle N\sqrt{dt}$.

TIA