Say $E$ is a smooth vector bundle, then is it necessarily true that $E$ has smooth sections that aren't constantly zero, namely $\Gamma^\infty(E) \neq \{0\}$ for $0$ the zero section of $E$.
My thinking is that perhaps we could look to extend local sections to global ones. In particular, if $\sigma$ is a local section of $E$ over some open set $U \subset B$, then we can find a smooth global section $\sigma'$ and an open neighbourhood $U'$ of $p$ in $B$ where the restriction of $\sigma$ to $U\cap U'$ is exactly the restriction of $\sigma'$ to $U\cap U'$. I'm unsure on how to prove this however, along with extending this to show that $\Gamma^\infty(E) \neq \{0\}$. Would the use of the smooth Urysohn's lemma be useful?