Thermodynamic limit of the first Lee-Yang zero

We complete the verification of the 1952 Yang and Lee proposal that thermodynamic singularities are exactly the limits in $\mathbb{R}$ of finite-volume singularities in $\mathbb{C}$. For the Ising model defined on a finite $\mathrm{\Lambda }\subset {\mathbb{Z}}^d$ at inverse temperature $\beta \ge 0$ and external field $h$, let ${\alpha }_1(\mathrm{\Lambda },\beta )$ be the modulus of the first zero (that closest to the origin) of its partition function (in the variable $h$). We prove that ${\alpha }_1(\mathrm{\Lambda },\beta )$ decreases to ${\alpha }_1({\mathbb{Z}}^d,\beta )$ as $\mathrm{\Lambda }$ increases to ${\mathbb{Z}}^d$ where ${\alpha }_1({\mathbb{Z}}^d,\beta )\in [0,\mathrm{\infty })$ is the radius of the largest disk centered at the origin in which the free energy in the thermodynamic limit is analytic. We also note that ${\alpha }_1({\mathbb{Z}}^d,\beta )$ is strictly positive if and only if $\beta$ is strictly less than the critical inverse temperature.