  By the Lindeberg principle, we develop in this paper an approximation to one
dimensional (possibly) asymmetric $\alpha$-stable distributions with $\alpha
\in (0,2)$ in the smooth Wasserstein distance. It is the first time that the
general stable central limit theorem is proved by the Lindeberg principle, and
that this theorem with $\alpha \in (0,1]$ is proved by a new method other than
Fourier analysis. Our main tools are a Taylor-like expansion and a Kolmogorov
forward equation.
