real analysis - Length convergence paradox

I saw this paradox a long time ago but have never been able to find a resolution to it. In the diagram below, the length of the straight line is of course $\sqrt{2}$, but the length of the 'pyramid' curve is always $2\cdot n\cdot\frac{1}{n}= 2$ (shown below is $n=10$). In the limit $n \to\infty$ it appears that the lines converge, but of course their lengths do not. Why does this happen, and how can this apparent paradox be resolved?