When you have your analytical solution beforehand, then you can use that by superimposing it on the way the simulation results behave when you increase the simulation size. A very common method is shown in this article which explains how to use an
R package called
ESGtoolkit, used for generating economic scenarios. The main idea would look something like this:
Like I said in the comment though, you should use images like this to show that it is the Monte Carlo method who will indeed converge to the true, analytical value, and not the other way around, i.e. "the MC method seems to converge around this number so that must be the true value".
When you don't know the analytic formula beforehand (because it might not even exist), the principle is the same but you might want to do a few slight modifications. An example would be the case of European Asian options, who do not have a closed formula for their price, and a common approach is the typical Monte Carlo simulations, but I found that if you were to perform only one "batch" of simulations, starting from say $N=25,000$ and simply increasing that $N$, it was a bit difficult to see the convergence (or be convinced that there is a convergence if you didn't know about the theory beforehand), so instead I did several batches of simulations for each simulation size value $N$, and plotted the resulting "price" of those on the same x-axis. As expected, if you keep the amount of batches the same but increase the simulation size $N$, the range of results starts decreasing:
Perhaps my point would have been clearer if I were to increase $N$ even more, but the running times started getting quite high already (and I wrote the code without targeting optimal performance).
Edit: For your $\alpha$ problem, I can think of two ways to go about it.
1. You either superimpose the individual lines that show the convergence on top of each other for each $\alpha$ and give them a different color according to each value of $\alpha$ that you test, or, if there are two many values of $\alpha$ and/or the simulation lines are too close to each other,
2. You create a 3d plot as Horst suggests, where sample size $N$ would be on the x-axis, $\alpha$ on the y-axis, and your value for $X$ on the z-axis. The axes or the label of the graph notwithstanding, it would look something like this.