(I checked later and the solution key writes that does have infimum Context matters… on exams you should ask for clarification when a question seems ambiguous. In the future I'll be careful to use the same conventions as the class.)
Question (from class): Why do we care about the Archimedian property?
Answer 1: Because we (mathematicians) want to understand the properties that characterize the real number system , and it turns out this property is not implied by the other properties we normally write down.
Answer 2: Because after realizing (or strongly suspecting) that the Archimedian property is not logically implied by the other properties of the reals, I may be curious to define a number system that looks relatively familiar (admitting addition, multiplication, inverses, linear ordering, etc.) but does not satisfy the Archimedian property. At some point you may invent the hyperreals. So, clearly articulating something familiar prepares us to investigate things that are less familiar.