  The Markov chain approximation of a one-dimensional symmetric diffusion is
investigated in this paper. Given an irreducible reflecting diffusion on a
closed interval with scale function $s$ and speed measure $m$, the
approximating Markov chains are constructed explicitly through the trace of the
Dirichlet form corresponding to the diffusion. One feature of our approach is
that it does not require uniform ellipticity on diffusion coefficient of the
limit object or uniform regularity on conductances of the approximative Markov
chains, as imposed usually in the previous related works.
