Since the early nineties, coalgebra has become an active area of research in which one tries to understand all kinds of infinite data types, automata, transition systems and dynamical systems from a unifying perspective. The focus of coalgebra is on observable behaviour and one uses coinduction as a central methodology.
In this lecture, we shall first summarize the key ingredients of the coalgebraic method, explaining how they largely derive from the theory of categories. As an extensive example, we shall then use coinduction to give new proofs of the theorems by Moessner (1951) and Paasche (1952), which are two non-trivial and entertaining combinatorial observations about infinite sequences (streams) of natural numbers. As we shall see, the heart of the matter in the coinductive approach is the identification of the circularity in the streams at hand, which then leads to surprisingly simple end elementary proofs.