My professor said that probability of 2 identical events in a very short amount of time (dt converges to 0) is 0. However, I did not agree with him about this. Is there a proof for that assertion? Intuition and thinking theoretically is a good thing, but it cannot be true if it is not supported by theorem which must be proved.
I tried to prove this theorem by letting the density function of an event is f(x,t). Then taking derivative, but yields no result.
Thank you very much for your help !!!
You might be referring to Poisson processes. In these models, when the intensity is $\lambda$, the probability that at least one event occurs in the time interval $(0,t)$ is $p_1(t)=1-\mathrm e^{-\lambda t}$. Thus, $p_1(t)\to0$ when $t\to0$.
However, what you have in mind might be the following. The probability that at least two events occur in the time interval $(0,t)$ is $p_2(t)=1-\mathrm e^{-\lambda t}-\lambda t\mathrm e^{-\lambda t}$. Thus, $p_2(t)\to0$ when $t\to0$ (this was obvious from the start since $p_2(t)\leqslant p_1(t)$) but, more precisely, $p_2(t)\ll p_1(t)$ since $p_1(t)\sim\lambda t$ and $p_2(t)\sim\frac12\lambda^2 t^2$ when $t\to0$.
In this sense, on a short time interval, two or more events are infinitely less likely than one event.