A lot of elementery (many would say easy) proofs undergraduates of mathematics encounter, are mostly straight forward (like substituing definitions) but very often contain one lightning strike. For example, showing that $f(x) = x^2$ is continous (in the sense of $\varepsilon - \delta$…) is straight forward except at one point: you have to choose (for a given $\varepsilon$) a special $\delta$ which solves the inequality.
So my motivation is to understand better where these $\delta$‘s, that often fall from heaven, really come from.