Solution
= Solution
In a finite interval, the effect at the origin of a fixed boundary spin a distance $N$ away is bounded by its two-point correlation, $(\tanh\beta)^N$. With two boundaries,
$$
|\langle\sigma_0\rangle_{\Lambda_N,\beta,0}^{\pm}|
\leq2(\tanh\beta)^N\longrightarrow0.
$$
Thus the plus and minus boundary limits coincide with the unique spin-flip-symmetric one-dimensional Gibbs state, and $\langle\sigma_0\rangle_{\beta,0}^{\mathbb Z,\pm}=0$ for every finite $\beta$.