We analyze interviews with 24 post-secondary students as they reason about infinite processes in the context of the tricky Tennis Ball Problem. By metaphorically projecting various properties from the finite states such as counting and indexing, participants envisioned widely varying final states for the infinite process. Depending on which properties they attended to, some participants recognized contradictions arising from these infinite projections and sought to resolve the contradictions in non-dismissive ways. The findings indicate that increasing their meta-level awareness of the features of their mental constructions may help students overcome common difficulties with the infinite. (Contains 1 figure.)