I am trying to understand why this inequality is true: $$\frac{\sum\limits_{i=1}^{k} x_i}{\sum\limits_{i=1}^{k} y_i} \le \max_i\{\frac{x_i}{y_i}\}$$. where $x_i,y_i \ge0$
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$\begingroup$ So, what exactly do you understand about each of these things already? $\endgroup$– NijCommented May 3, 2021 at 9:14
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1 Answer
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Rewrite it as:
$$\sum\limits_{i=1}^{k} x_i \le \max_i\{\frac{x_i}{y_i}\}\sum\limits_{i=1}^{k} y_i$$
Then, let $M=\max_i\{\frac{x_i}{y_i}\}$, and we have $My_i \le \max_i\{x_i\}$, so that:
$$\sum\limits_{i=1}^{k} x_i \le \sum\limits_{i=1}^{k}\max_i x_i$$
which is true.