However, what we will do is not let Δt approach zero, but just keep it very small. In this way we get a broken line, consisting of little straight line segments, that is an approximation to the true solution. This is called “Euler’s Method”

There is a theorem (Shadowing Lemma) that guarantees that as the time step Δt approaches 0, the blue broken line of Euler’s approximation approaches a red curve.