Meaning of $d(x_m,x_n)\to 0$ on $\mathbb{R}$

Consider the next lemma:

Lemma. A sequence $(x_n)_{n\in\mathbb{N}}$ on a metric space $(X,d)$ is a Cauchy sequences, if and only if, $d(x_m,x_n)\to 0$ on $\mathbb{R}$.

I don't understand the formal meaning of the expression "$d(x_m,x_n)\to 0$ on $\mathbb{R}$". Intuitively, it means that, when $m$ and $n$ are large, the distance from $x_m$ to $x_n$ is zero, i.e.$$\lim_{m\to\infty}\lim_{n\to\infty} d(x_m,x_n)=0.\hspace{1 in} (1)$$I know what it means for a sequence to converge, but $\lim_{n\to\infty} d(x_m,x_n)$ is not a sequence as far as I understand, so the expression (1) has no meaning. What's the sequence $d(x_m,x_n)$ corresponds? What's the formal definition of "$d(x_m,x_n)\to 0$ on $\mathbb{R}$"?