I was reading Guillemin and Pollack's book to self-study differential topology, where I encountered the following two related problems (Exercise 5 and 6 in Section 2.3):
(Exercise 5) Assume that one compact submanifold $X \subset Y$ intersects another sub-manifold $Z$ and satisfies $\dim X + \dim Z < Y$. Show that $X$ can be pulled away from $Z$ by an arbitrarily small deformation. That is to say, for any $\epsilon>0$ there exists a deformation $X_t = i_{t}(X)$ such that $X_1 = i_1(X)$ doesn't intersect $Z$ and $|x-i_1(x)|<\epsilon \ ( \forall \ x \in X)$
(Exercise 6) The claim in Exercise 5 above can be sharpened as follows: if $Z$ is closed in $Y$ and $U$ is an open set in $X$ containing $Z \cap X$, then the deformation $X_t = i_t(X)$ above can be chosen to be constant outside of $U$.
A related post about Exercise 5 is here: Show that $X$ can be pulled apart from Z by an arbitrary small deformation
I have been stuck on these two problems for several weeks. Basically, I have tried following the hint in the book and Prof. Shifrin's answer to solve the first problem, but I'm still quite confused. It seems that we need to define some smooth map $F:X \times S \rightarrow Y$ to apply the Transversality Theorem, where $S$ is some ball of radius $\epsilon$, right? However, how can we ensure that the whole map $F$ and its boundary restriction $\partial F$ are transversal to $Z$? For the second problem, I'm still having no clear clue...
I would appreciate it if someone is willing to provide some hints/suggestions, as it's my first time reading this book. Thanks again in advance!