Let $M=(\mathbb{R}^n, g)$ be a Riemannian manifold equipped with the Riemannian metric $g= \psi g'$ where $g'$ is the standard Riemannian metric on $\mathbb{R}^n$ and $\psi(x)=e^{x_n}$ for $x=(x_1, \dots, x_n) \in \mathbb{R}^n$. Show that $M$ is not complete.
I tried to use the Hopf–Rinow theorem and show that as a metric space $M$ is not complete, but I don't know how this should be done. When considering $M$ as a metric space we define $$d(p,q)=\inf_\gamma \ell(\gamma)$$ to be the metric where $\gamma$ is an admissible path from $p$ to $q$.
Now in order for this to not be a complete metric space one needs to find a Cauchy sequence in $M$ that does not converge. This isn't very clear to me how we should define these kinds of sequences and what would it mean to compute $d(p_n, p_m)$ say if the sequence is $(p_n)_n \in M$? Are we taking admissible paths between these points and trying to minimize the distance?
Some geodesics in $M$ are easy to find almost without calculation. The maps ('reflections') $F_i:M\to M$, $$F_i(x_1,\dots,x_n)=(x_1,\dots,x_{i-1}, -x_i, x_{i+1},\dots,x_n)$$ are isometries for $1\leq i\leq n-1$. Let us take a point $P\in M$ and a vector $v\in T_PM$ such that both $P$ and $v$ are fixed by all the $F_i$'s, e.g. $P=(0,\dots, 0)$ and $v=\partial_{x_n}$. Then the geodesics $\gamma$ given by $\gamma(0)=P$, $\gamma'(0)=v$ must be fixed by all the $F_i$'s, and so it is of the form $\gamma(t)=(0,\dots,0,f(t))$ for some function $f$ s.t. $f(0)=0$ and $f'(0)=1$. Finally, since for every $t$ we have $g(\gamma'(t),\gamma'(t))=g(v,v)=1$, we get $$e^f (f')^2 =1,$$ therefore $f(t)=2\log(1+t/2)$. This geodesic is evidently incomplete.