The problem and solution are attached as TheProblemAndSolution.jpg.
Definition of the limit of a sequence.
The Attempt at a Solution
I understand how P = ϵ + |A| can be seen as an upper bound that proves that the sequence is bounded, but for the last bit of the solution where the solution says “we choose P as the largest one of the numbers [. . .]”, why does it say a_N instead of a_n, shortly after? Is that a typo? Also, there isn't necessarily a value of a_n that is the largest (such that no value of a_n will ever reach ϵ + |A|), unless the author(s) meant to say that ϵ + |A| is one of the numbers to choose from such that ϵ + |A| is the largest number mentioned by the solution.
Could someone please clarify the last part of the solution for me?
I know this is a pedantic request, but I would greatly appreciate any input!