Let $f:\mathbb{R^2} \to \mathbb{R^2}$ whose derivative is nonsingular.let D denote the open unit ball $\{v:|v|<1\}$. show that $D \subset f(D)$

Question

let $f:\mathbb{R^2} \to \mathbb{R^2}$ smooth function whose derivative at every point is nonsingular. suppose that $f(O)=O$ and for all $v \in R^2$ with $|v|=1$ , $|f(v)| \geq 1$. let D denote the open unit ball $\{v:|v|<1\}$. show that $D \subset f(D)$ (hint: show that $f(D) \cap D$ is closed in D)

by Inverse mapping theorem $f(D) \cap D$ is open in $D$. What to do next? Any help is appreciated.