Why does decay of correlations imply absence of order?

In a few articles I have read, a two-point correlation function $\langle g(x)g(y) \rangle$ is shown to decay with increasing distance of $x$ and $y$, and this is then taken to imply an absence of the corresponding order. For example $\langle g(x)\rangle$=0 in the thermodynamic limit.

A concrete example: lets consider a heisenberg model on a square lattice with N sites: $$H=\sum_{x,y}J_{x,y}\ \vec{S_x}\cdot\vec{S_y} + h\sum_x S^3_x$$ why does

$$\lim_{N \to \infty}|\langle S_x^3 S_y^3\rangle|≤e^{-|x-y|}$$

imply that $\lim_{N \to\infty} \frac{1}{N}\sum_x \langle S_x^3\rangle = 0$?

Is it important how fast the correlation function decays? Would the mean magnetisation also disappear in the case of a polynomial decay of the correlation function? Does the result depend on the dimension of the lattice?