An elementary proof of the existence and uniqueness of solutions to an initial value problem

In this note, we show a classical result on the local existence and uniqueness of a solution to an initial value problem subject to a Lipschitz condition. We use only elementary tools from mathematical analysis, without involving any integration. We proceed by showing that the Cauchy iterates converge on a dense subset of the interval and subsequently proving that the extension of this limit function to the whole interval is a solution to the Cauchy problem.