Given two sequences of positive numbers $a_1, a_2, \dots a_k$ and $b_1, b_2, \dots b_k$. Prove that $$ \min_{i} \frac{a_i}{b_i} \leq \frac{\sum_{i}a_i}{\sum_{i}b_i} \leq \max_{i} \frac{a_i}{b_i} $$

I have seen this inequality used in various proofs but I have no idea how to start proving such an inequality. Some hints would be useful.


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Set $M = \max_{i} a_i/b_i$. This means that for every $i$ $a_i/b_i \leq M$, so $a_i \leq Mb_i$. From this you get $$ \sum_{i} a_i \leq \sum_{i} Mb_i = M \sum_{i} b_i, $$ which is the same as $$ \frac{\sum_{i} a_i}{\sum_{i} b_i} \leq M = \max_{i} \frac{a_i}{b_i}. $$

For the other side you can do an analogous proof.

  • $\begingroup$ Very elegant. Thanks Hugo! $\endgroup$ – user1105 Mar 4 '16 at 15:09
  • $\begingroup$ Does this inequality have a name or can someone tell me a text in which it appears? I would like to be able to cite it by name or a reference in which it appears. $\endgroup$ – D_J_S Feb 13 at 23:23

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