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I have proved and am using the following simple lemma in my current research problem:

Let $\{a_1,...,a_m\}$ and $\{b_1,b_2,...,b_n\}$ be set of positive numbers such that $\sum_{i=1}^m a_i < \sum_{j=1}^n b_j$. Then there exist positive numbers $\epsilon_1,...,\epsilon_n$ such that $b_j - \epsilon_j > 0 \,\forall\, 1\le j \le n$ and $\sum_{i=1}^n \epsilon_i = \sum_{j=1}^{m} a_j $.

I have proved this using elementary euclidean geometry. I have the following questions.

a) is this problem (or any of its generalisations!) known is the literature? if so kindly share with me so that simply I can cite them in my paper rather than giving the proof again.

b) If not (I strongly believe that this is not the case), which branch of Mathematics or papers in which area I should read in order to understand and improve this lemma to more general setup.

Thanks for your valuable time.

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    $\begingroup$ I assume you mean $b_i-\epsilon_i>0$. In a paper I would say this is sufficiently obvious not to require proof, but note that you can just let $b_i-\epsilon_i = b_i(\sum a_i)/(\sum b_i)$. $\endgroup$ – Simon L Rydin Myerson May 24 '18 at 8:28
  • $\begingroup$ Hi. Thanks. I got your point. But I think you meant $\epsilon_i = b_i(\sum a_i)/(\sum b_i)$ ? $\endgroup$ – GA316 May 24 '18 at 9:24
  • $\begingroup$ Yes indeed, that's right. $\endgroup$ – Simon L Rydin Myerson May 24 '18 at 12:39

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