In lectures on the h-cobordism theorem, Milnor writes of an isotopy $h_t:\mathbb{R}^n\to\mathbb{R}^n$, with $\mathbb{R}^n=\mathbb{R}^a\oplus\mathbb{R}^b$ in his notation, the following lemma:
The first item of the lemma is clear to me (there's a formula for $h_t$ in the previous page and it's easy to work out the inequality as he says).
The second item of the lemma on the other hand doesn't need to use the formula of $h_t$ at all, only that it maps $0$ to $0$ and transversality, according to the last sentence of his proof. Would someone have a hint at why?
First, I should say the condition 2) is abusing language in the sense that the inequality is not for all $x\in\mathbb{R}^a$, but rather in $\mathbb{R}^a\cap \{\text{neighborhood of the origin in }\mathbb{R}^n \text{ Milnor refers to}\}$.
Inside a closed small disk where the only intersection point is the origin, if we can never find a minimal bound for $|\pi_ah_t(x)|/|x|$, then there is a sequence $x_n$ converging to the origin such that $|\pi_ah_t(x_n)|/|x_n|$ goes to zero.
By transversality, $d(\pi_a\circ h_t\vert_{\mathbb{R}^a})(0)$ should be a matrix with full rank. Thus in particular $$\lim_{x_n\to 0}\dfrac{||\pi_a\circ h_t(x_n) - \pi_a\circ h_t(0)||}{||x_n-0||}$$ should not be zero, since this would imply that $d(\pi_a\circ h_t)(0) = 0$. This shows we indeed have a minimum $k$, as stated in the Lemma.