How does one prove the following inequality is true?

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?

• Using your previous result and induction might work – vrugtehagel May 16 '17 at 14:27
• 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. – Macavity May 16 '17 at 14:29
• Induction. Only concern is that the series converges. – fleablood May 16 '17 at 15:13
• By "any pair of sets," do you mean to include infinite sets? – Connor Harris May 16 '17 at 15:17
• @ConnorHarris nope, only finite ones. – Max Muller May 16 '17 at 21:23