Let $F: X \times [0, 1] \to Y$ be a homotopy such that for any $t \in [0,1]$ the map $F( \cdot, t) : X \to Y$ is proper. Is it true in general that $F$ is proper? 


I am interested in particular in the case when $X$ and $Y$ are both complete metric spaces. 


Any help will be very much appreciated!


EDIT: Just to be clear, by *proper map* I mean a map such that the preimage of a compact set is a compact set.



Proper homotopy
The following counterexample is stolen from page 1 of [this paper](https://arxiv.org/abs/1808.08073) by Thomas Rot. The map $[0,1]\times\mathbb R\to\mathbb R$ given by $(t,x)\to (1-t)x^2+x$ is not proper, e.g., the preimage of $\{0\}$ contains the sequence $(1-\frac{1}{n}, -n)\_{n\in\mathbb N}$. Of course, each parabola $x\to (1-t)x^2+x$ is proper.


