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I am thinking if the following condition is in general true: $\frac{n}{m} \leq \frac{\sum_{i = 1} ^ {n} a_i}{\sum_{i = 1} ^ {m} b_i}$, when $n \leq m$ and $a_i \geq 0$, $b_i \geq 0$ but i cannot find a proof.

Can you please help me with that?

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    Why on earth would that be true? Replacing $a_i$ by $\epsilon a_i$ for $\epsilon>0$, you can make the right hand side as close to zero as you might wish.2012-09-14
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    Or what about $a_i=1, b_i = 2$? You might be able to salvage your conjecture by imposing some additional conditions on the $a$s and $b$s, but as it stands, it's just wrong.2012-09-14

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