# Proof of Mermin-Wagner Theorem

There are many presentations of the proof of the Mermin-Wagner theorem in many different contexts (which talk about quantum vs. classical, existence of unique Gibbs measure or non-zero mean magnetization, talk about a Lie group of symmetries or just the XY model). I am aware of the relations between the various statements of the theorem (correlation inequalities e.g.) but:

I am looking for the simplest, most immediate and direct proof one could find for the following statement, which I guess is slightly stronger than the statement about mean magnetization:

Theorem: If $$S:\{1,\dots,L\}^2\to\mathbb{S}^{N-1}$$ is spin field of the $$O(N)$$ model ($$N\geq2$$), whose partition function is defined as $$Z_L = \prod_{x\in\{1,\dots,L\}^2}\int_{S_x\in\mathbb{S}^{N-1}}\mathrm{d}\mu(S_x)\exp(\beta\sum_{x,y\in\{1,\dots,L\}^2:x\sim y}S_x\cdot S_y)$$ where $$\mu$$ is the uniform volume measure on $$\mathbb{S}^{N-1}$$ then $$\lim_{x\to\infty}\lim_{L\to\infty}\mathbb{E}_L[S_x\cdot S_0] = 0\qquad(\beta > 0)\,.$$

The best is if you could prove (or sketch a proof) in a paragraph or few right here, otherwise if there is a great reference for this it would also be appreciated. Just as an example, googling the term, the first result is Scholarpedia and I think one has to work a bit to get from what they state to the statement above. It would be nice to have something direct.