Partial Converse 2 in Guillemin Pollack (Page 24) This might be a really stupid question, but in Guillemin Pollack's partial converse 2 in Chapter 1, it starts with an arbitrary submanifold $Z$ in $X$, and then says , for some open set $W \in X$, we have an immersion $Z \rightarrow W$. Why is this true in general? 
 A: For future reference, it would be helpful if you could give a more precise description of what the book says, and which part gives you trouble. I happen to have the book handy, but many readers of this site will not. 
What the book says is a little different from what you quoted. "Partial Converse 2" on page 24 says, in part,

Every submanifold of $X$ is locally cut out by independent
  functions. ... More specifically, let $Z$ be a submanifold of
  codimension $l$, and let $z$ be any point of $Z$. Then we claim that
  there exist $l$ independent functions $g_1,\dots,g_l$ defined on some
  open neighborhood $W$ of $z$ in $X$ such that $Z\cap W$ is the common vanishing set of the $g_i$. (Here we are stating the converse to the proposition for the submanifold $Z\cap W$ in the manifold $W$.) This converse follows immediately from the Local Immersion Theorem for the immersion $Z\to W$; ...."

This makes it clear that they've made an error. Note the words "open neighborhood $W$ of $z$ in $X$." So when they introduce $W$, they're not claiming that $Z$ is contained in $W$; instead, they're just claiming that $W$ is some neighborhood of $z$ such that $Z\cap W$ is immersed in $W$. So when they write "the Local Immersion Theorem for the immersion $Z\to W$," what they really should have written is "... for the immersion $(Z\cap W) \to W$."
EDIT: To address the question of why the inclusion of a submanifold into an open subset is an immersion, note that if $Z$ is a submanifold of $X$, then the inclusion map $\iota:Z\to X$ is an immersion, because it's locally a diffeomorphism onto its image and therefore $d\iota_z$ is injective for each $z\in Z$. Since this just depends on what happens in a neighborhood of each point $z$, it follows that if $W\subseteq Z$ is any open set, then the restricted map $\iota|_{Z\cap W}\colon Z\cap W\to W$ is still an immersion.
