I can prove that if $\frac{a_{i}}{b_{i}} \geq \frac{a_{j}}{b_{j}} $, it follows that $\frac{a_{i}}{b_{i}} \geq \frac{a_{i} + a_{j}}{b_{i} + b_{j}} $. However, I'd like to generalise this result by proving that if $$ \frac{\sum_{i \in S} a_{i}}{\sum_{i \in S} b_{i}} \geq \frac{\sum_{i \in T} a_{i}}{\sum_{i \in T} b_{i}},$$ it follows that $$ \frac{\sum_{i \in S} a_{i}}{\sum_{i \in S} b_{i}} \geq \frac{\sum_{i \in S \cup T} a_{i}}{\sum_{i \in S \cup T} b_{i}} $$ for any pair of sets $S$ and $T$.

Do you know how to prove the latter statement?

  • $\begingroup$ Using your previous result and induction might work $\endgroup$ – vrugtehagel May 16 '17 at 14:27
  • $\begingroup$ The first statement requires $b_j>0$. The second is easily proved considering $\sum_S a_i =A$ and so on directly from the first. $\endgroup$ – Macavity May 16 '17 at 14:29
  • $\begingroup$ Induction. Only concern is that the series converges. $\endgroup$ – fleablood May 16 '17 at 15:13
  • $\begingroup$ By "any pair of sets," do you mean to include infinite sets? $\endgroup$ – Connor Harris May 16 '17 at 15:17
  • $\begingroup$ @ConnorHarris nope, only finite ones. $\endgroup$ – Max Muller May 16 '17 at 21:23

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