Working Out Some Math

An Error in A7:

\frac{(\frac{\lambda^2\sigma}{4})^{\frac{1}{2}}}{\sigma^{\frac{1}{2}}}=\frac{\sum^n_{1}(\delta^{z}\bar{g}_{ij}^{z})}{\sum^n_{1}(\alpha^{z}\bar{g}_{ij}^{z})} \implies \frac{\lambda}{2}=\frac{\sum^n_{1}(\delta^{z}\bar{g}_{ij}^{z})}{\sum^n_{1}(\alpha^{z}\bar{g}_{ij}^{z})}

𝜌: Correlation between genetic nurture effects and direct genetic effects

Since 𝜌 is a correlation, then -1 \leq \rho \leq 1.

1\geq\rho \implies 1\geq\rho^2\implies\lambda^2\geq\rho^2\lambda^2\implies \frac{\lambda^2}{4}\geq\frac{\lambda^2\rho^2}{4}\implies 1+\lambda\rho+\frac{\lambda^2}{4}\geq 1+\lambda\rho+\frac{\lambda^2\rho^2}{4}\implies (1+\lambda\rho+\frac{\lambda^2}{4})^{\frac{1}{2}}\geq (1+\lambda\rho+\frac{\lambda^2\rho^2}{4})^{\frac{1}{2}}\implies \frac{1}{(1+\lambda\rho+\frac{\lambda^2}{4})^{\frac{1}{2}}}\leq\frac{1}{(1+\lambda\rho+\frac{\lambda^2\rho^2}{4})^{\frac{1}{2}}}\implies \frac{1+\frac{\lambda\rho}{2}}{(1+\lambda\rho+\frac{\lambda^2}{4})^{\frac{1}{2}}}\leq\frac{1+\frac{\lambda\rho}{2}}{(1+\lambda\rho+\frac{\lambda^2\rho^2}{4})^{\frac{1}{2}}}\implies\frac{1+\frac{\lambda\rho}{2}}{(1+\lambda\rho+\frac{\lambda^2}{4})^{\frac{1}{2}}}\leq\frac{1+\frac{\lambda\rho}{2}}{((1+\frac{\lambda\rho}{2})^2)^{\frac{1}{2}}}\implies \frac{1+\frac{\lambda\rho}{2}}{(1+\lambda\rho+\frac{\lambda^2}{4})^{\frac{1}{2}}}\leq1