  In this paper, we study nonlinear stability of the 3D plane Couette flow
$(y,0,0)$ at high Reynolds number ${Re}$ in a finite channel $\mathbb{T}\times
[-1,1]\times \mathbb{T}$. It is well known that the plane Couette flow is
linearly stable for any Reynolds number. However, it could become nonlinearly
unstable and transition to turbulence for small but finite perturbations at
high Reynolds number. This is so-called Sommerfeld paradox. One resolution of
this paradox is to study the transition threshold problem, which is concerned
with how much disturbance will lead to the instability of the flow and the
dependence of disturbance on the Reynolds number. This work shows that if the
initial velocity $v_0$ satisfies $\|v_0-(y,0,0)\|_{H^2}\le c_0{Re}^{-1}$ for
some $c_0>0$ independent of $Re$, then the solution of the 3D Navier-Stokes
equations is global in time and does not transition away from the Couette flow
in the $L^\infty$ sense, and rapidly converges to a streak solution for $t\gg
Re^{\frac 13}$ due to the mixing-enhanced dissipation effect. This result
confirms the transition threshold conjecture proposed by Trefethen et
al.(Science, 261(1993), 578-584). To this end, we develop the resolvent
estimate method to establish the space-time estimates for the full linearized
Navier-Stokes system around the flow $(V(t,y,z), 0,0)$, where $V(t,y,z)$ is a
small perturbation(but independent of $Re$) of the Couette flow $y$.
