Restriction of a Lie bracket on the space of section of a vector bundle..

Question

Let $A\longrightarrow M$ be a vector bundle and $U\subseteq M$ an open set. Suppose I have a lie bracket on $\Gamma(A)$ such that if $\rho:A\longrightarrow TM$ is a bundle map then $$[a, fb]=f[a, b]+\rho(a)(f)b,$$ for all $a, b\in \Gamma(A)$ and $f\in C^\infty(M)$. I want to show this lie bracket restricts to a Lie bracket $[, ]_U$ on $$\Gamma(U):=\{\gamma|_U: \gamma\in \Gamma(A)\}.$$ I read that it suffices showing that if $a, b\in \Gamma(A)$ vanishes on $U$ then $[a, b]$ also vanish on $U$. I have already shown this but I don't understand why it is sufficient. Can anyone help me? Thanks