For multi-objective optimization problem. A method to find weakly efficient is using max-ordering, which is $min_{x\in X}\ max_{i=1,...n}f_i(x)$.
How to prove that:An optimal solution $x^o$ of the max-ordering problem $min_{x\in X}\ max_{i=1,...n}f_i(x)$ is weakly efficient.
I have try to prove by contradiction. Firstly suppose that there is a point $x^c$ that $f_j(x^c) < f_j(x^o)$ . When the $f_j(x^c) = max_{i=1,...n}f_i(x)$. It is contradict with the suppose.
However, when the $f_j(x^c) \neq max_{i=1,...n}f_i(x)$ . There is seems not contradiction.
Where is the problem of my proof?