In this paper, we theoretically study the proportional fair (PF) scheduler in the context of ultra-dense networks (UDNs). Analytical results are obtained for the coverage probability and the area spectral efficiency (ASE) performance of dense small cell networks (SCNs) with the PF scheduler employed at base stations (BSs). The key point of our analysis is that the typical user is no longer a random user as assumed in most studies in the literature. Instead, a user with the maximum PF metric is chosen by its serving BS as the typical user. By comparing the previous results of the round-robin (RR) scheduler with our new results of the PF scheduler, we quantify the loss of the multi-user diversity of the PF scheduler with the network densification, which casts a new look at the role of the PF scheduler in UDNs. Our conclusion is that the RR scheduler should be used in UDNs to simplify the radio resource management (RRM).