Suppose $X\to M$ is a $C^\infty{}$-fibration. In his book Partial Differential Relation, Gromov claims (pg. 121) that this can be realized as a $C^\infty{}$-sub-fibration of the trivial bundle $M\times \mathbb{R}^m \to M$, for some large enough $m$. I fail to see how this is possible.
For the simpler case of a locally trivial fiber bundle $F\hookrightarrow X \to M$, my idea was to embed $F$ in an Euclidean space by Whitney and then use local triviality of the fiber bundle. But I got stuck defining the map on the total space.
Could someone please give me some idea or any reference for this? Any help is appreciated.