How to prove that a system is real-time?
It depends on your definition of "real-time system".  A standard definition is: a system is real-time if it will respond within $T$ seconds, with probability at least $P$.  Here you need to fill in the parameters $T,P$; e.g., it might be $T = 1$ millisecond and $P = 0.999$.

How do you prove that this condition is met?  One way is to find a way to generate random workloads, from the same distribution as the system will be exposed to in practice, and count how often the system meets the deadline and how often it doesn't.  Another way is to prove a theorem (e.g., with formal methods) about the maximum time it might take to respond; see the literature on worst-case execution time analysis.  The latter might be particularly challenging in many cases.

There's an entire literature on this topic, that you can read more about.