Consider a discrete stationary process $\{X_n\}$ with autocorrelation $\rho_k=0,\forall k\ge2$, prove that $|\rho_1|\le\frac12$. WLOG assume that $EX_n=0,Var(X_n)=1$.
I have tried to use Cauchy's inequality, $\rho_1=EX_1X_2=E[X_1(X_2-\rho_1X_3+\rho_1^2X_4-\dots)]\le\sqrt{EX_1^2E(X_2-\rho_1X_3+\rho_1^2X_4-\dots)^2}$. This leads to $|\rho_1|\le\sqrt{\frac{3-\sqrt{5}}{2}}$, which is not enough.
Due to the stationary process $ \{X_n\} $ with autocorrelation $ \rho_k=0$, $\forall k\ge 2$, hence $X=\{X_n\} $ is an MA(1) and with following expression \begin{equation*} X_n= a\epsilon_n+b\epsilon_{n-1}, \end{equation*} where $ \{\epsilon_n\} $ is a white noise sequence WN$(0,\sigma^2)$ ($\mathsf{E}[\epsilon_n]=0$, $\mathsf{E}[\epsilon_m\epsilon_n] = \delta_{mn}\sigma^2$ ).
Now calculating the $\rho_1$ of $X$ get \begin{equation*} \rho_1=\frac{\mathsf{E}[X_0X_1]}{\mathsf{E}[X_0^2]}=\frac{ab}{a^2+b^2}. \end{equation*} Hence, $ |\rho_1|\le\frac12 $.