Working on a problem in Lee's Intro to Smooth Manifolds (2nd edition, problem 8, chapter 6). The problem states:
Prove that a proper continuous map between smooth manifolds is homotopic to a proper smooth map between said manifolds.
Let $M$ and $N$ be the manifolds with $F:M \to N$ a proper continuous map. Since $F$ is continuous we automatically have the Whitney Approximation theorem: we can embed $N \subseteq \mathbb{R}^n$ for some $n$ and that $F$ is $\delta$-close to a smooth map $\tilde{F}: M \to \mathbb{R}^n$. Because $\tilde{F}(M)$ can fit into some tubular neighborhood $U \subseteq \mathbb{R}^n$ of $N$, we have there is a mapping $r: U \to N$ which is a retraction and a submersion into $N$. We can then take our homotopy to be $H: M\times [0,1] \to N$ given by
$$ H(x,t) \;\; =\;\; r \left ( (1-t) F(x) + t\tilde{F}(x) \right ). $$
The theorem shows that $F$ is homotopic to a smooth map $r\circ \tilde{F}$, but we want to show that $r\circ\tilde{F}$ is proper.
I'm somewhat lost as for proof strategies and need some help. Lee hints that we need to show that $\tilde{F}$ is proper, and hopefully $r$ can easily shown to be proper. Where I'm getting stuck is in picking an arbitrary $K \subseteq N$ which is compact, how do we show $\tilde{F}^{-1}(K)$ is compact as well? I'm thinking we can pick a sequence in $\tilde{F}^{-1}(K)$ and find a convergent subsequence. Somewhere in here I'm assuming we need to use the fact that $||F(x) - \tilde{F}(x)|| < \delta(x)$ for all $x \in M$, given $\delta:M \to \mathbb{R}_+$.
Any and all help is appreciated on this. Thanks in advance!