$\mathbb{R}$-bundle
Solution 1:
Since it is a differential geometry question suppose that $B$ is paracompact, let $(U_i)_{i\in I}$ a trivialisation, and $f_i$ a partition of unity subordinate to $(U_i)$. Let $h_i:U_1\rightarrow\mathbb{R}$ defined by $h_i(x)=1$, define $h=\sum_i f_ih_i$ since $f_i\geq 0, h_i>0$.