<0$. There exists a unique $\alpha_0\in (\alpha_0,\gamma_0)$. Set $\beta_0:=\gamma_0-\alpha_0+k-0=\alpha_0+\beta_0-\alpha_0$ and $\gamma_0:=\alpha_0+\beta_0-k+0=\gamma_0$. Hence $$\label{step0} \alpha_0<\alpha_0<\gamma_0\quad\text{and}\quad 0<\beta_0<\beta_0\quad\text{and}\quad \gamma_0=\gamma_0\,.$$ 