Is there an $L<1$ such that $\left\Vert f(z_1) -f(z_2) \right\Vert_2 \leq L \left\Vert z_1 -z_2 \right\Vert_2 $

Question

Let $f(x,y)=(\sqrt{4-y^2},\sqrt{1-\frac{x^2}{16}})$. I'm trying to show if there exists an $L<1$ such that

$\left\Vert f(z_1) -f(z_2) \right\Vert_2 \leq L \left\Vert z_1 -z_2 \right\Vert_2 $ for all $z_1,z_2 \in S=\{0,2\} \times \{0,2\}$.

I'm stuck, any tips?

Answer 1

You know that if such an $L$ were to exist it would provide a bound on the Jacobian matrix. Now consider the Jacobian close to $y=2$.