On the continuity of the time constant of first-passage percolation

Let U be the distribution function of the non-negative passage time of an individual edge of the square lattice, and let a 0n be the minimal passage time from (0, 0) to (n, 0). The process a 0n /n converges in probability as n → ∞to a finite constant μ ( U ) called the time constant. It is proven that μ ( U k ) → μ ( U ) whenever U k converges weakly to U .