Suppose we are given an infinite list of positive, pairwise-prime integers $n_1, n_2, n_3,\cdots$ and a second infinite list of integers $a_1, a_2, a_3, \cdots$. Can we find a solution to the following system of congruences:

(1)We could try to apply CRT to this problem, but we would run into problems with our system being infinite: we would get an infinite lcm as our modulus and our solution would be an infinite sum.

We can alternatively try an inspection method: if there is a solution $X$, then there must be some $N$ such that for all $i \geq N$, we get $X \equiv a_i \mod n_i$. Thus, we could try to look far ahead enough into the system to find such a solution. Aside from the fact that this is rather infeasible, the following system has no solution:

(2)Using this method, we would look at this and say "ooo, the solution must be 2" and be wrong. Thus, we could try to truncate the system after finding such a bound and apply CRT to it, then check the solution it gives against the supposed solution.