  We study the long-time asymptotics of the probability P_t that the
Riemann-Liouville fractional Brownian motion with Hurst index H does not escape
from a fixed interval [-L,L] up to time t. We show that for any H \in ]0,1],
for both subdiffusion and superdiffusion regimes, this probability obeys
\ln(P_t) \sim - t^{2 H}/L^2, i.e. may decay slower than exponential
(subdiffusion) or faster than exponential (superdiffusion). This implies that
survival probability S_t of particles undergoing fractional Brownian motion in
a one-dimensional system with randomly placed traps follows \ln(S_t) \sim -
n^{2/3} t^{2H/3} as t \to \infty, where n is the mean density of traps.
