  Magnetization measurements were performed to investigate the critical
behavior of the field-induced magnetic ordering in gapped spin system
TlCuCl$_3$. The critical density of the magnons was obtained as a function of
temperature and the magnon-magnon interaction constant was evaluated. The
experimental phase boundary for $T < 5$ K agrees almost perfectly with the
magnon BEC theory based on the Hartree-Fock approximation with realistic
dispersion relations. The phase boundary can be described by the power law
$[H_{N}(T)-H_{c}] propto T^{phi}$. With decreasing fitting temperature range,
the critical exponent ${phi}$ decreases and converges at $phi_{BEC} =3/2$
predicted by the magnon BEC theory.
