Asymptotic results cannot be proven by computer simulation, because they are statements involving the concept of infinity. But we should be able to obtain a sense that things do indeed march the way theory tells us.

Consider the theoretical result $$\lim_{n\rightarrow\infty}P(|X_n|>\epsilon) = 0, \qquad \epsilon >0$$

where $X_n$ is a function of $n$ random variables, say identically and independently distributed. This says that $X_n$ converges in probability to zero. The archetypal example here I guess is the case where $X_n$ is the sample mean minus the common expected value of the i.i.d. r.v.'s of the sample,

$$X_n = \frac 1n\sum_{i=1}^nY_i - E[Y_1]$$

QUESTION: How could we convincingly show to somebody that the above relation "materializes in the real world", by using computer simulation results from necessarily finite samples?

Please do note that I specifically chose convergence to a constant.

I provide below my approach as an answer, and I hope for better ones.