Let $S\subset M$ be a properly embedded submanifold (in particular, $S\subset M$ is a closed submanifold) and let $V$ be a smooth vector field on $M$ tangent to $S$. Since we have that $M\setminus S\subset M$ is open, we know that it is a submanifold of $M$ using the subspace charts. Does this imply that $V$ is tangent to $M\setminus S$?
Any ideas?
Edit: Here's an idea I had. $V$ is a smooth vector field on $M$, then $V_p\in T_pM$ for each $p\in M$. Then since $M\setminus S\subset M$ is open, the map $d\iota_p^{-1}:T_pM\to T_p(M\setminus S)$ is an isomorphism for $\iota:M\setminus S\to M$ the inclusion, so $d\iota_p^{-1}(V_p)\in T_p(M\setminus S)$. Then consider $d\iota^{-1}(V)$. Then this is a vector field (?) which is $\iota$ related to $V$, so $V$ is tangent to $M\setminus S$.