  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.
