Proper maps and transversality I'll begin with the question, which is intrinsically interesting:

Let M be a manifold with some submanifold Y. Suppose that $W \rightarrow M$ is a smooth, proper map. Does there exist another map $W \rightarrow M$ homotopic to the original that is ALSO proper and transverse to the submanifold Y?

Let me note that I am definitely not assuming the manifolds are compact. 
Why I care: This question came up while thinking about the geometric description of complex cobordism given by Quillen in "Elementary proofs of some results of cobordism theory using Steenrod operations." I have a geometric description of the coboundary map in the Mayer-Vietoris sequence but as of now it relies on the answer to the above question being "yes."
 A: The comments explain how to prove the fact.
If you want to put a formal wrapping around it, consider the strong (Whitney) $C^\infty$ topology on the space of maps $W\to M$. The strong $C^0$ topology on $C(M,\mathbb R)$ can be defined as follows: for every function $h:M\to\mathbb R$ which is positive and locally bounded away from 0 (but may tend to 0 at infinity), declare the set 
$$U_h:=\{f\in C^0(M): \forall x\in M \ |f(x)|<h(x)\}$$
a neighborhood of zero; this gives you a prebase of the topology. For smooth maps between manifolds the definition is similar but involves derivatives and a locally finite covering by charts (or, alternatively, a complete Riemannian metric, on which the resulting topology does not depend).
In the strong $C^\infty$ topology, the set of proper maps is open, and the set of maps transverse to $Y$ is open and dense. For a reference, see e.g. Hirsch, "Differential topology" (1976), Chapter 2 and Theorem 2.1(a) in Chapter 3.
A: Regarding the coboundary map in complex cobordism, I think that this was done by Dold in "Geometric Cobordism and the Fixed Point Transfer" (at least for oriented cobordism) in 2.10.
