Is $\mathbb{N}$ a complete metric space with this metric $d(a,b) = \sqrt{1-2\frac{\gcd(a,b)}{a+b}}$?

Question

Is $X:=\mathbb{N}$ a complete metric space with this metric $d(a,b) = \sqrt{1-2\frac{\gcd(a,b)}{a+b}}$?

Thanks for your help!

Edit: This metric plays a role in the formulation of the abc-conjecture:

https://mathoverflow.net/questions/352054/the-abc-conjecture-as-an-inequality-for-inner-products

Answer 1

If $a\ne b$, then letting \begin{align*} a_1&=\frac{a}{\gcd(a,b)}\\ b_1&=\frac{b}{\gcd(a,b)}\\ \end{align*} we have $$ \frac{\gcd(a,b)}{a+b} = \frac{1}{a_1+b_1} \le \frac{1}{3} $$ so $d(a,b) \ge \sqrt{1-2{\,\cdot\,}\frac{1}{3}}={\large{\frac{1}{\sqrt{3}}}}$.

It follows that every Cauchy sequence must be eventually constant, hence convergent.

Therefore $X$ is complete.