  For a given K-polystable Fano variety $X$ and a natural number $l$ such that
$(X, \frac{1}{l} B)$ is log canonical for some $B\in |-lK_X|$, we show that
there exists a rational number $0<c_1<1$ depending only on $X$ and $l$, such
that $D\in |-lK_X|$ is GIT-(semi/poly)stable under the action of Aut(X) if and
only if the pair $(X, \frac{\epsilon}{l}D)$ is K-(semi/poly)stable for some
rational $0<\epsilon<c_1$.
