📂Lemmas

Summarizing Inequalities in the Form of an Inequality

Theorem

Given positive numbers $x_{1} , \cdots , x_{n}$ and $a_{1} , \cdots , a_{n} > 0$, and constant $\theta \in \mathbb{R}$. $$\forall i \in [n] : x_{i} < a_{i} \theta \iff \max_{i \in [n]} {{ x_{i} } \over { a_{i} }} < \theta$$

Theorem

For all $(\implies)$, that $i \in [n]$ satisfies $x_{i} / a_{i} < \theta$ implies that even the greatest $x_{i} / a_{i}$ is less than $\theta$. That the greatest $x_{i} / a_{i}$ is less than $\theta$ implies that $i \in [n]$ satisfies $x_{i} / a_{i} < \theta$ for all $(\implies)$.

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Explanation

In the Opposite Direction

Necessary for the proof of the theorem related to Sufficient Statistics. Naturally, the following theorem can be considered in the opposite direction. $$\forall i \in [n] : x_{i} > b_{i} \theta \iff \min_{i \in [n]} {{ x_{i} } \over { b_{i} }} > \theta$$