  We investigate the convergence of the Galerkin approximation for the
stochastic Navier-Stokes equations in an open bounded domain $\mathcal{O}$ with
the non-slip boundary condition. We prove that
  \begin{equation*}
  \mathbb{E} \left[ \sup_{t \in [0,T]} \phi_1(\lVert (u(t)-u^n(t))
  \rVert^2_V) \right] \rightarrow 0
  \end{equation*} as $n \rightarrow \infty$ for any deterministic time $T > 0$
and for a specified moment function $\phi_1(x)$ where $u^n(t,x)$ denotes the
Galerkin approximation of the solution $u(t,x)$. Also, we provide a result on
uniform boundedness of the moment $\mathbb{E} [ \sup_{t \in [0,T]} \phi(\lVert
u(t) \rVert^2_V) ] $ where $\phi$ grows as a single logarithm at infinity.
Finally, we summarize results on convergence of the Galerkin approximation up
to a deterministic time $T$ when the $V$-norm is replaced by the $H$-norm.
