    \begin{align*}
        f_i(x)&+\varepsilon_1+2L\varepsilon_2-f_i(\bar{x})=f_i(x_i,x^*_{-i}+b^*_{-i})+\varepsilon_1+2L \varepsilon_2-f_i(x^*+b^*)\\
        &=f_i(x_i,x^*_{-i}+b^*_{-i})-f_i(x_i,x^*_{-i})+L  \varepsilon_2+f_i(x^*)-f_i(x^*+b^*) + L \varepsilon_2+f_i(x_i,x^*_{-i})-f_i(x^*)+\varepsilon_1 \geq 0,
    \end{align*}
