Borel's lemma

Borel's lemma is a result in the theory of functions, particularly in the context of real or complex analysis. It states the following: Let \( f \) be a function defined on an open interval \( I \subseteq \mathbb{R} \) that is infinitely differentiable (i.e., \( f \in C^\infty(I) \)). If \( f \) and all of its derivatives vanish at a point \( a \in I \) (i.e.