We prove two versions of the celebrated theorem of Picard: the local and the global existence and uniqueness results for differential equations. We use Banach’s principle as a main tool in our analysis, and this makes us realize that it is the completeness of the space of continuous functions that is the reason for the existence of solutions to differential equations. In the mean time we get acquainted with the notion of equivalent metrics and learn that, in proving the existence of a fixed point of a map, it is sometimes more convenient to use one norm and sometimes another, equivalent one.