Summary: A Poisson process \({P}_{\lambda}\) on \(\mathbb{R}^{d}\) with causal structure inherited from the the usual Minkowski metric on \(\mathbb{R}^{d}\) has a normalised discrete causal distance \({D}_{\lambda}(x,y)\) given by the height of the longest causal chain normalised by \({\lambda}^{1/d}{c}_{d}\). We prove that \({P}_{\lambda}\) restricted to a compact set \(Q\) converges in probability in the sense of J. Noldus [Classical Quantum Gravity 21, No. 4, 839–850 (2004; Zbl 1055.83034)] to \(Q\) with the Minkowski metric.