In this study we consider a classic paradox of infinity and its variations and suggest how the sources of misleading intuition can be analysed using the concept of uniform convergence of functions. We then examine how six mathematics honour students engage with a variation of the paradox. Despite their advanced mathematical training, the participants experienced considerable difficulty in addressing the presented paradoxical situation. Drawing on data from written responses and individual interviews, we describe the students' approaches in an attempt to reconcile their intuitive perceptions with their mathematical computations.