Is this proof correct or am I trying to prove a falsehood?

Question

Suppose that we have three possible outcomes $\alpha$, $\beta$ and $\gamma$ (if that helps, suppose they are monetary amounts) and some threshold $x$. Then, we also know that \begin{gather*} x\geqslant\alpha\\ x\geqslant\tau\beta+(1-\tau)\gamma \end{gather*} for any $\tau\in[0,1]$. My goal is to show that if the above Inequalities hold, then \begin{gather*} x\geqslant(1-\rho-\pi)\alpha+\rho\beta+\pi\gamma \end{gather*} for any $(\rho,\pi)\in[0,1]^2$ such that $\rho+\tau\in[0,1]$. My attempt is as follows. Let's go step by step.

1. Given the premises, define some $\varphi\in[0,1]$ and then write \begin{gather*} x\geqslant\varphi\alpha+(1-\varphi)\left[\tau\beta+(1-\tau)\gamma\right] \end{gather*}

2. Then, resolve some multiplications to get \begin{gather*} x\geqslant\varphi\alpha+(\tau-\varphi\tau)\beta+(1-\varphi\tau-\varphi-\tau)\gamma \end{gather*}

3. Because $\tau\in[0,1]$ and $\varphi\in[0,1]$, it follows that \begin{gather*} 0\leqslant\tau-\varphi\tau\leqslant1\\ 0\leqslant1-\varphi\tau-\varphi-\tau\leqslant1 \end{gather*}

4. Therefore, let $\rho=\tau-\varphi\tau$ and $\pi=1-\varphi\tau-\varphi-\tau$ and then write: \begin{gather*} x\geqslant\varphi\alpha+\rho\beta+\tau\gamma \end{gather*}

5. Then, notice that \begin{align*} 1-\rho-\pi&=1-\tau+\varphi\tau-1-\varphi\tau+\varphi+\tau\\ &=1+\varphi-1\\ &=\varphi \end{align*}

6. And voilà! It seems we're done, because \begin{gather*} x\geqslant(1-\rho-\pi)\alpha+\rho\beta+\pi\gamma \end{gather*}

Therefore, my question is: is this simple proof correct? If it is false, please state why; if it is true only under some unstated conditions, please state them. If it is correct... Well, then just say it.

Thank you all very much in advanced for your time and effort.

Answer 1

Plugging in $\tau = 1$ we get $x \ge \beta$, while plugging in $\tau = 0$ we get $x \ge \gamma$. Let $\delta = \max\{\alpha,\beta,\gamma\}$; then we have $x \ge \delta$. So, we have: \begin{align*} (1-\rho-\pi)\alpha + \rho\beta + \pi\gamma &\le (1 - \rho - \pi) \delta + \rho \delta + \pi \delta\\ &\le\delta\\ &\le x. \end{align*}