Diameter bound for a fixed-density binomial random graph ($\mathrm{Pr}(\mathrm{diam}G(n,p)\le 2)\to 1$)

For fixed $0<p<1$, the missing common-neighbour count in a binomial random graph has expectation tending to zero. The Markov inequality therefore shows that every pair has a common neighbour with probability tending to one. On that event every graph distance is at most two, so the graph is connected and its graph diameter is at most two.