Let $M=\mathbb{R}\times\mathbb{S}^{n-1}$ be equipped with the warped product metric $g(x,\theta)=dx^2+e^{-x^2}h_\theta$ , where $h_\theta$ is standard metric on $\mathbb{S}^{n-1}$ . Prove that $(M,g)$ is a complete Riemannian manifold .
Using Hopf-Rinow , it is enough to prove metrical completeness . So we choose two points $x,y\in M$ . Denote $x=(x_1,\theta_1)$ and $y=(x_2,\theta_2)$ and $d_g$ as geodesic distance in $(M,g)$ . Then clearly $$d_g(x,y)=|x_1-x_2|+d_h(\theta_1,\theta_2)\geq|x_1-x_2|$$ But I'm not able to show the reverse inequality in order to show $M$ is metrically equivalent to $\mathbb{R}$ , which is complete .
Any help is appreciated .