  We revisit a result of Uchiyama (1980): given that a certain integral test is
satisfied, the rate of the probability that Brownian motion remains below the
moving boundary $f$ is asymptotically the same as for the constant boundary.
The integral test for $f$ is also necessary in some sense. After Uchiyama's
result, a number of different proofs appeared simplifying the original
arguments, which strongly rely on some known identities for Brownian motion. In
particular, Novikov (1996) gives an elementary proof in the case of an
increasing boundary. Here, we provide an elementary, half-page proof for the
case of a decreasing boundary. Further, we identify that the integral test is
related to a repulsion effect of the three-dimensional Bessel process. Our
proof gives some hope to be generalized to other processes such as FBM.
