= Solution
Assume short-range ferromagnetic exchange $J>0$ and zero external field. In the one-dimensional <Ising model>, a <domain wall> costs only $2J$, independent of the lengths of the domains it separates. There are order-$N$ positions for one wall, so its positional entropy offsets any finite energy cost at nonzero temperature as the system grows. Domain walls consequently have a positive density at every $T>0$, destroying a uniform spontaneous <spin magnetization>. Periodic boundaries require an even number of walls, but that constraint does not remove their nonzero thermodynamic-limit density.
The <One-dimensional Ising correlation function> makes this argument explicit. A bond is broken with probability $p_w=e^{-2\beta_{\rm th}J}/(1+e^{-2\beta_{\rm th}J})$, and the infinite-chain correlation is
$$
G(r)=(1-2p_w)^{r/a}=(\tanh\beta_{\rm th}J)^{r/a},\qquad
\xi=\frac{a}{-\log\tanh\beta_{\rm th}J}<\infty\quad(T>0).
$$
The dominant eigenvalue of the <transfer matrix for the one-dimensional Ising model> is $2\cosh\beta_{\rm th}J$, so the free energy per spin $-k_BT\log[2\cosh\beta_{\rm th}J]$ is analytic for $T>0$. \b[There is no finite-temperature transition in one dimension; only the zero-temperature limit orders.]
In two dimensions a reversed cluster is enclosed by a <Peierls contour>. A contour of length $\ell$ costs $2J\ell$, while the number of such contours surrounding a fixed site grows at most as a polynomial in $\ell$ times $3^\ell$: after the initial edge, a nonbacktracking square-lattice walk has at most three choices per step. With plus boundary conditions, flipping the interior of a specified contour yields the Boltzmann suppression $e^{-2\beta_{\rm th}J\ell}$. Therefore
$$
\Pr(\sigma_0=-1)\leq\sum_{\ell\geq4}C\ell^2\bigl(3e^{-2\beta_{\rm th}J}\bigr)^\ell.
$$
For sufficiently low temperature this bound is below $1/2$, giving a positive spontaneous magnetization. This is the energy-entropy mechanism of the <Peierls argument>.
At sufficiently high temperature, the graphical expansion bounds correlations by sums of connecting paths weighted by $(\tanh\beta_{\rm th}J)^\ell$. The same nonbacktracking count converges geometrically when $3\tanh\beta_{\rm th}J<1$, giving exponential decay and a disordered phase. Since an ordered low-temperature phase and a disordered high-temperature phase both occur, \b[the two-dimensional model has a finite-temperature phase transition.] The argument establishes its existence and the dimensional distinction, not the exact value of its critical temperature.
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