The real numbers form the basis for most of secondary school mathematics, from analytic geometry to calculus and normal distributions. However, both in secondary school and in undergraduate mathematics, it is customary to stay with an informal description of what the real numbers are, while the university courses on analysis certainly stipulate that they form a complete metric space, and prove many advanced theorems based on that. Future secondary school teachers typically see little relation between the advanced theory they learnt from undergraduate mathematics courses and what they are going to teach in secondary school. The general “gap” is often called “Klein’s double discontinuity”. This chapter focuses on the particular case of real numbers. More concretely, we focus on identifying certain challenges related to the second discontinuity and the efforts and effects of trying to bridge it through a “capstone course” at the University of Copenhagen. We analyse students’ work with assignments designed to enable them to draw on university mathematics to solve problems involving real numbers and software commonly used in secondary school. This analysis is complemented by a survey and some interviews to get more insight into students’ perceptions about the relevance of the real number theme for high school teaching.