Is $\mathbb{R}\setminus\{0\}$ endowed with $d(x,y)=|\frac{1}{x}-\frac{1}{y}|$ complete?

Question

Consider $\mathbb{R}^*=\mathbb{R}\setminus\{0\}$ and let $d:\mathbb{R}^{*}\times\mathbb{R}^*\to [0,+\infty[$ the function defined as $d(x,y)=|\frac{1}{x}-\frac{1}{y}|$.

Is the metric space $(\mathbb{R}^*,d)$ complete?

If I have the following sequence $\{n\}_{n\in\mathbb{N}}$ it does not converge once $\lim_{n\to\infty} \frac{1}{n}=0$ but $0\notin\mathbb{R}^*$

I have the above idea but I do not think is enough.

Question:

Can someone provide a proof for this statement?

Is $(\mathbb{R},d)$ complete?

Thanks in advance!