Existence of critical exponents for the 3D Ising model

Nearest-neighbour Ising model on $Z^3$ at $\beta_c$. Prove that $\langle s_0 s_x\rangle_{\beta_c}=\left|x\right|^{-(1+\eta)+o(1)}$ and that the correlation length $\xi(\beta)=(\beta_c-\beta)^{-\nu+o(1)}$, with numerical values $\eta\ \mathrm{approximately}\ 0.0363$ and $\nu\ \mathrm{approximately}\ 0.6300$. Answer: a proof of existence of the limits, with or without their values.

A proof that the exponents $\eta$ and $\nu$ exist as limits.