We now show that this implies that τ − τ′ tends to 0 in probability. From the definition of μ(m,n) in Equation 5, we have μ(m,n) < 4. Pick a constant (say 10) >4. Since as m → ∞, (L − μ(m,n))/σ(m,n) tends to TW in distribution, and σ(m,n) → 0, it follows that P(L > 10) tends to 0 as m → ∞. Similarly, P(T < 1/2) tends to 0. Take ɛ > 0. From Equation 29:
