This manuscript focuses on how students make sense of proofs. Participants were students who engaged in peer-review conferences of each other's attempted proofs in a graduate-level real analysis course for mathematics teachers. Building on the concept of distance from conversational analysis, we distinguish how three types of distance (epistemic, rhetorical, and ontological) between a student and a particular claim influence sensemaking. This article also explores the impact of students' sensemaking on their perceptions of proof.