Formal proof on how many weapon levels can be neglected.

Let us take k = 1 and denote the stopping time of obtaining a weapon of cost i as . Then

independent

We can bound from below the geometric distriubtion by uniform one, in a sense that

independent

where is exactly an event in which we eventually obtain a weapon of level x. Now let us present the probability space of Y₁, .., Y_x as Ω = [0, 1] × [0, 2] × .. × [0, x] which is a hypercuboid with volume x!, therefore probability measure P has density in lebesgue measure. It is easy to see that is contained in a hypercone of lebesgue volume . By taking into consideration the density g, we get an upper bound

Now back to the original task. Let us denote the total profit from selling the weapons as T ≤ n². We want the following expectation to be small:

For n = 10⁵ and ε = 10^- 9, using the formula above we can check that x = 900 is enough.