Let $F:\mathbb{R}^2\to\mathbb{R}^2$ be defined by $F(x,y)=(x+2y+x^2\ ,\ y-x^3+y^2)$. Then show that for $p_0=(4,1)$ and $p_1=(1,1)$ there exists $\delta>0$ such that for every $\vec y\in B(p_0,\delta)$ there exists $\vec x\in B(p_1,\delta)$ such that $F(\vec x)=\vec y$.
I think it has to be a pretty straightforward application of the local inversion theorem; since $DF(p_1)$ is invertible it is applicable. However, I'm not able to show that for the same $\delta$ neighborhoods of the two points the restriction of $F$ can still be assumed to be surjective; if we were allowed to chose two independent $\delta_1$ and $\delta_2$ then it would be straightforward. How to argue?
Edit:
Picking just the smaller of the two $\delta_1,\delta_2$ doesn't work immediately because if the restriction $F:B(p_1;\delta_1)\to B(p_0;\delta_2)$ is surjective and $\delta_1>\delta_2$ then $F:B(p_1;\delta_2)\to B(p_0;\delta_2)$ is not forced to be surjective.