This paper reports a teaching experiment in which two students engaged in tasks that challenged them to describe a final state for a variety of infinite iterative processes. The results from the study indicate that the students used multiple reasoning strategies for addressing these tasks. Refinements in the students' reasoning occurred as students constructed relationships between the problems they were solving and problems they had solved previously, applying some of the reasoning strategies that they used for one problem to make sense of or solve another problem. We discuss how these findings relate to the existing body of research on infinite iterative processes.