I have smooth maps $f,h:\mathbb{R}\rightarrow (0,\infty )$ with $f(t)\geq k$ and $h(t)\geq \frac{1}{\mid t\mid}$ for all $\mid t\mid >c$ for some$k,c>0$.
I want to prove that $$g=f(t)ds^2+h(t)dt^2$$ is a complete metric on $\mathbb{R}^2$, where $(s,t)$ are standard coordinates.
I suspect that I might need the result (which I have proven) that if $g'$ and $g$ are Riemannian metrics on a Riemannian manifold $M$, with the property that $g(X,X)\geq g'(X,X)$ for all $X\in T_pM$, for all $p\in M$ and $(M,g')$ is complete, then $(M,g)$ is complete.
My guess would be to use the fact that $(\mathbb{R}^2,g')$, with $g'$ the usual metric on $\mathbb{R}$, is complete. Then I would need to show that $g(X,X)\geq g'(X,X)$ for all $X\in T_p\mathbb{R}^2$, for all $p\in\mathbb{R}$. But I can't seem to get anywhere with this idea. Any help is much appreciated.