It happens regularly when, surprise, you deal with infinities in some other place.
A good, fairly intuitive example is integration where, I think, it's common to prove convergence using AoC in the multidimensional case.
More or less, you want to integrate a function over, say, a square. We'll do it with a generalization of the Riemann sum by arbitrarily dividing up that square into little squares, measuring the function at one place in each square, multiplying the areas by those measurements, and summing.
Then we take that process to its infinite limit. We have to be smart here, but we end up with an infinite set of contiguous pieces of our original square. Or, an infinite set of sets. We'd like to measure our function once from each of those pieces, so we need to choose one point from each piece. Which is easily dispatched with AoC.
(In 1-dimensional integration, each subdivision of the domain has a total ordering, so you don't necessarily need AoC. Technically my example of a square also has a total ordering, so we could get away with not using AoC, but as you start to twist coordinates in the domain space more and more you can imagine places where seeking out an ordering of the space might be challenging. But no matter, AoC still works!)
A good, fairly intuitive example is integration where, I think, it's common to prove convergence using AoC in the multidimensional case.
More or less, you want to integrate a function over, say, a square. We'll do it with a generalization of the Riemann sum by arbitrarily dividing up that square into little squares, measuring the function at one place in each square, multiplying the areas by those measurements, and summing.
Then we take that process to its infinite limit. We have to be smart here, but we end up with an infinite set of contiguous pieces of our original square. Or, an infinite set of sets. We'd like to measure our function once from each of those pieces, so we need to choose one point from each piece. Which is easily dispatched with AoC.
(In 1-dimensional integration, each subdivision of the domain has a total ordering, so you don't necessarily need AoC. Technically my example of a square also has a total ordering, so we could get away with not using AoC, but as you start to twist coordinates in the domain space more and more you can imagine places where seeking out an ordering of the space might be challenging. But no matter, AoC still works!)