Not your ordinary sum

For any bijection $f \,:\, \mathbb{R} \to \mathbb{R}$ , the function $d(x,y) \; = \; \big|f(x) - f(y)\big|$ is a metric. Given the function $f(x) \; =\; \left\{ \begin{array}{lll} x^{-1} & \hspace{1cm} & x > 0 \\ x & & x \le 0 \end{array}\right.$ Then $(\mathbb{R},d)$ is a complete metric space and we deduce that $d\left(\sum_{r=1}^n r,0\right) \; = \; \left(\sum_{r=1}^nr\right)^{-1}$ so that $\lim_{n \to \infty} \sum_{r=1}^n r \; = \; 0$ with respect to this metric.