Let $G$ be a Lie group, and let $v: G \rightarrow T(G)$ be a left invariant vector field. Then $v$ is uniquely determined by its value at $e = 1_G$, so $v \mapsto v(e)$ defines an injection of left invariant vector fields into $T_e(G) = \mathfrak g$. I'm trying to show this correspondence is surjective.
Given $X \in \mathfrak g$, the corresponding $v$ should be defined by the formula $v(x) = T_e(\ell_x)(X)$, where $\ell_x: G \rightarrow G$ denotes left translation.
The first thing I'm trying to check is that $v$ is smooth. Should this be obvious? I think I have a way of showing arguing this (in my answer below), but I would be interested in knowing whether there is a simpler way.