No, that is not what the problem involves. The problem, rather, involves showing that the whole journey (not just part of it) can be completed in a finite time. That is what the problem is about.
The paradox comes about because we have made things complicated by dividing Homer’s journey into an infinite number of parts so that it appears Homer will have an infinite number of steps to take and may never complete his journey. Fortunately mathematics shows us that the sum of an infinite number of parts can be finite (as it is in this case) and thus resolves the problem we have created for ourselves.