Lower and Upper bounds for Ratio of Sum of two Sequences of positive numbers [closed]

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.

closed as off-topic by Namaste, Davide Giraudo, Daniel W. Farlow, C. Falcon, JMPApr 3 '17 at 0:18

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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.

• Very elegant. Thanks Hugo! – user1105 Mar 4 '16 at 15:09
• 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. – D_J_S Feb 13 at 23:23