In the book Semi-Riemannian Geometry with applications to Relativity by Barrett O'Neill, in Chapter 2 (Tensors), he stated that:
However, I don't get why he had to use a bump function in the proof. Wouldn't it be correct if he just said that since $X^i(p) = 0$ then the RHS (without bump function) will be zero?
I see that he imposed a bump function $f$ to say that $fX^i$ is smooth and $f \partial _i \in \frak{X} \mathrm{(M)}$. But isn't $\partial _i \in \frak{X} \mathrm{(M)}$ already?
What would fail in the proof had there been no bump function?