A difficulty in understanding the solution of problem 1.7.17 in Guillemin and Pollack.(p.47)

Question

The question is given below:

And here is exercise (16):

And here is the solution to exercise(17)

But I have difficulties in understanding the following parts of the solution:

1-Why the codomain of the defined $h$ in the second line is $\mathbb{R}$?

2-Why $h$ is clearly smooth as stated in the third line?

3-Why we are using U x {0}, what is the importance of using the singleton $0$?

4-Why K x {0} is compact? and why this leads to that $h > 2\delta$ for some $\delta > 0$?

5-And by which property of continuity of h, there exists an open set $U'$ such that $h > \delta$ on $U'$?

6-When usually Tube lemma is applied, when we need what, we apply it?

Thank you!

Answer 1

1. Because $\det(Hf_t|_x)^2 + \sum_{i=1}^k \frac{\partial f_t}{\partial x_i}$ is a real number.
2. Presumably, there is a blanket assumption of smoothness for "homotopic families of functions" in this text. In general, this wording is not sufficient to suppose that $f_t$ is differentiable at all, much less smooth. However, since the solution immediately takes derivatives without any mention of conditions, I'm guessing that the authors have already placed a blanket restriction on such families to be be smooth. Differentiating, summing and multiplying smooth functions results in smooth functions, so $h$ would also be smooth.
3. Because it is given in the question that $f_0$ is Morse. That is the starting point of their logic: when $t = 0$.
4. Because $K$ is compact. And because $h$ is continuous on a compact set, and therefore must have a minimum.
5. $h^{-1}((\delta, \infty))$ is open.
6. I have no idea what you are asking. Please clarify.