The French railroad metric on $\mathbb{R}^2$ is defined as $||x - y||$ is $x, y$ are on a line that goes through the origin, or otherwise as $||x|| + ||y||$.
Prove this forms a complete metric space.
Note: This question is discussed elsewhere. I am posting nonetheless because the proof there is debated (with 10 comments going back and forth), and I'd like help verifying what to me seems like a simple proof.
Proof: If $x$ and $y$ are on the same line through the origin, refer to them as corail. Corail forms a partition of $\mathbb{R}^2 -$ the origin.
Let $\{S_n\}$ be a Cauchy sequence in this space. If $S$ eventually restricts to a single line through the origin, the metric reduces to Euclidean metric on the real line, and is therefore complete.
Alternatively, if $S$ always has points $a, b$ not corail, $d(a,b) = ||a|| + ||b|| \geq max(||a||,||b||)$. Since $S$ is Cauchy, $d(a,b) \to 0$, so $||a|| \to 0$ and $||b|| \to 0$. A similar argument applies to any point $c$ other than the origin appearing in the sequence after $a$ and $b$: $c$ is either not corail with $a$ or not corail with $b$. Thus, all points in $S$ tend to an Euclidean norm of $0$, and so $S$ clearly converges to the origin.
Is this proof correct?