I need some hints to prove the following lemma (John Lee's $\textit{Introduction to Smooth Manifolds 2nd Ed}$ p.201) :
EXTENSION LEMMA FOR VECTOR FIELDS ON SUBMANIFOLDS: Suppose $M$ is a smooth manifold and $S\subseteq M$ is an embedded submanifold. Given a smooth vector field $X$ on $S$ show that there is a smooth vector field $Y$ on a neighborhood of $S$ in $M$ such that $Y=X$ on $S$. Show that every such vector field extends to all of $M$ if and only if $S$ is properly embedded.