Roth's theorem states that
Let $\alpha$ be a real algebraic number of degree $\geq 3$. Then for every $\kappa >2$ there exists a constant $c(\alpha,\kappa)>0$ such that $$|\xi -\alpha| \geq c(\alpha,\kappa) H(\xi)^{-\kappa}$$ for every $\xi \in \mathbb{Q}$.
Prove that the following statement is equivalent to Roth's theorem:
Let $\alpha_1,...,\alpha_m$ be distinct real algebraic numbers of degree $\geq 3$. Then for every $\kappa >2$ there is a constant $c>0$ such that $$\prod_{i=1}^m |\alpha_i -\xi| \geq cH(\xi)^{-\kappa}$$ for $\xi \in \mathbb{Q}$
Let $|\alpha_t-\xi| =\min_{1 \leq i \leq m} |\alpha_i-\xi|$ then $$ \prod_{i=1}^m |\alpha_i -\xi| \geq \left(|\alpha_t-\xi| \right)^m \geq c(\alpha_t,\kappa)^m H(\xi)^{-m\kappa}$$ However, I cannot indicate $c(\alpha_t,\kappa)^m H(\xi)^{-(m+1)\kappa} \geq c$ for some constant $c$. Does anyone have other approach?