SeriesInformation ; Physical review letters 116(15), 151105(2016). doi:10.1103/PhysRevLett.116.151105 ; Abstract ; The Fermi Large Area Telescope (LAT) Collaboration has recently released a catalog of 360 sources detected above 50 GeV (2FHL). This catalog was obtained using 80 months of data re-processed with Pass 8, the newest event-level analysis, which significantly improves the acceptance and angular resolution of the instrument. Most of the 2FHL sources at high Galactic latitude are blazars. Using detailed Monte Carlo simulations, we measure, for the first time, the source count distribution, dN/dS, of extragalactic γ-ray sources at E>50 GeV and find that it is compatible with a Euclidean distribution down to the lowest measured source flux in the 2FHL (∼8×10$^{−12}$ ph cm$^{−2}$ s$^{−1}$). We employ a one-point photon fluctuation analysis to constrain the behavior of dN/dS below the source detection threshold. Overall, the source count distribution is constrained over three decades in flux and found compatible with a broken power law with a break flux, Sb, in the range [8×10$^{−12}$,1.5×10$^{−11}$] ph cm$^{−2}$ s$^{−1}$ and power-law indices below and above the break of $α_2$∈[1.60,1.75] and $α_1$=2.49±0.12, respectively. Integration of dN/dS shows that point sources account for at least 86$^{+16}_{−14}$% of the total extragalactic γ-ray background. The simple form of the derived source count distribution is consistent with a single population (i.e., blazars) dominating the source counts to the minimum flux explored by this analysis. We estimate the density of sources detectable in blind surveys that will be performed in the coming years by the Cherenkov Telescope Array. ; Other ; Published by APS, College Park, Md.