The rate of convergence in law of the maximum of an exponential sample

Summary  We derive a uniform rate of convergence of (1– n^-1x)ⁿ to e^-x(x < 0). It provides a uniform rate of convergence for the distribution of the largest order statistic in a sample from an exponential distribution to the "double exponential" extreme value distribution. It likewise provides a rate of convergence for the distribution of the smallest order statistic from a uniform distribution.