Metric coinduction is a form of coinduction that can be used to establish properties of objects constructed as a limit of finite approximations. One proves a coinduction step showing that some property is preserved by one step of the approximation process, then automatically infers by the coinduction principle that the property holds of the limit object. This can often be used to avoid complicated analytic arguments involving limits and convergence, replacing them with simpler algebraic arguments. This paper examines the application of this principle in a variety of areas, including infinite streams, Markov chains, Markov decision processes, and non-well-founded sets. These results point to the usefulness of coinduction as a general proof technique.