Solution
= Solution
In the one-dimensional zero-field <Ising model>, the bond variables $\tau_j=\sigma_j\sigma_{j+1}$ are independent and
$$
\mathbb E\tau_j=\frac{e^\beta-e^{-\beta}}{e^\beta+e^{-\beta}}=\tanh\beta.
$$
Since $\sigma_0\sigma_x=\prod_{j=0}^{x-1}\tau_j$ for $x>0$,
$$
\langle\sigma_0\sigma_x\rangle_{\beta,0}^{\mathbb Z}
=(\tanh\beta)^{|x|}
=e^{-c(\beta)|x|},
\qquad c(\beta)=-\log(\tanh\beta)>0.
$$