Bound on the running maximum of a random walk with small drift

We derive a lower bound for the probability that a random walk with i.i.d. increments and small negative drift mu exceeds the value x > 0 by time N. When the moment generating functions are bounded in an interval around the origin, this probability can be bounded below by 1 - O(x vertical bar mu vertical bar logN). The approach is elementary and does not use strong approximation theorems.