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|\mu| \log N). The approach is elementary and does not use strong approximation theorems.