Let $g_0$ be the usual Euclidean metric on $\mathbb{R}^{2}$ and define Riemannian metric $g=(1+{x_1}^2+x_2^2)^{2}g_0$, also on $\mathbb{R}^{2}$. Show that if $\alpha:[0,L]\rightarrow \mathbb{R}^2$ is curve, then its length is greater wrt metric g than $g_{0}$. Hence, by considering closed and bounded sets in $(\mathbb{R}^{2},g)$ and using Hopf-Rinow, show that $(\mathbb{R}^{2},g)$ is complete. (by complete I mean the exponential map is defined at every point on the manifold and for all tangent vectors at each point).
We're given $(\mathbb{R}^{2},g_0)$ is complete and I easily showed the length arugment. My struggle is with showing the latter. Since we're in $\mathbb{R}^{2}$ I wanted to use Heine-Borel but this would completely ignore the choice of metric and would contradict the next question which asks to show a $\mathbb{R}^{2}$ with a different metric is not complete.
Any suggestions are greatly appreciated.
