This is actually a good example of the difference between pointwise and uniform convergence. Consider the characteristic function of the vase at time t, g_t : N -> {0, 1}. Then g_t(n) = 1 if and only if ball n is in the vase at time t. It will actually help to convert g_t to a function on the real numbers, by making f_t(x) = g_t(floor(x)).

Now, for each ball n, there is a time when it will be destroyed, and therefore will never be in the vase after that time. So the characteristic function f_t(x) converges pointwise to f(x) = 0. This is presumably what you mean by the limit of the vase being the empty set.

But the criterion of uniform convergence is that for any epsilon>0 there is a t such that f_t is within epsilon of the limit everywhere. Which is obviously not true, because at any time t there are some balls in the vase, and so the characteristic function is 1 somewhere. So f_t(x) does not uniformly converge to anything.

As it happens, without uniform convergence, the limit of the integrals of f_t(x) (which just so happens to be the number of balls in the vase, by my setup) is not generally equal to the integral of the limiting function f(x). So, in a way it is not really true that you can say

in the limit as T approaches infinity, there are infinitely many balls in the vase

as the integral does not transfer to the limit.