In this study, the conceptions and knowledge of mathematics students and pre-service teachers for upper secondary school of the notion of convergence of sequences will be investigated. Empirical results of the second test, which will be applied after finishing the topic in class, are presented, and also empirical results concerning the correlation between prior knowledge collected by the first test instrument and knowledge after the treatment in class. With findings about (mis-)conceptions of students of an Analysis-1-classregarding the concept of convergence, a contribution will be made to the state of mathematics education research regarding the concept of limit. In addition, a learning environment for the guided reinvention of the concept of convergence will be developed in a design-based research study. For this purpose, a concept of Przenioslo (2005) will be adapted and analysed in a scientific study. Based on a given set of diverse examples and one non-example of convergent sequences, the students will reinvent the definition of the convergence of a sequence towards a limit. For possible problematic situations remedial help will be prepared. The reinvention processes from two implementations will be analysed retrospectively. The result of this design-based research study is a local instruction theory for the guided reinvention of the notion of convergence. Furthermore, by means of the test results an effect of the attendance at the workshop, although small, will be established.