I'm trying to prove this intuitive result by Kushnirenko, contained in this article of his:
https://iopscience.iop.org/article/10.1070/RM1967v022n05ABEH001225
Let $A=\{a_1,\dots,a_n\}$ be a set of non negative real numbers with $\sum_{i=1}^n a_i = 1$, and $X=\{x_1,\dots,x_n\}$ set of non negative real numbers such that $\sum_{i=1}^n a_i x_i < \delta$
The assertion is:
" For every $M>0$ large enough, there exists a subset of $A$, namely $\{a_{i_1},\dots, a_{i_k}\}$ with $\sum_{\ell=1}^k a_{i_\ell} \geq 1-1/M$ and a subset of $X$, namely $\{x_{i_1},\dots, x_{i_k}\}$ such that $x_{i_\ell}\leq M\delta$, for all $\ell=1,\dots,k$."
My attempt using induction on $n$:
"The case $n=1$ is valid, since $a=1$ and $x_1<\delta$.
Suppose validity for $n_0$, that is, given $A=\{a_1,\dots,a_{n_0}\}$ such that $\sum_{i=1}^{n_0} a_i = 1$ and $X=\{x_1,\dots,x_n\}$ such that $\sum_{i=1}^{n_0}a_i x_i < \delta$, for every $M>0$ large enough, there exists subsets of $A$ ,$\{a_{i_1},\dots, a_{i_k}\}$ with $\sum_{\ell=1}^k a_{i_\ell} \geq 1-1/M$ and subset of $X$ ,$\{x_{i_1},\dots, x_{i_k}\}$ such that $x_{i_\ell}\leq M\delta$, for all $\ell=1,\dots,k$.
Consider sets with $n_0+1$ elements $A'=\{a_1,\dots,a_{n_0+1}\}$ and $X'= \{x_1,\dots,x_{n_0}, x_{n_0+1}\}$ such that $\sum_{i=1}^{n_0+1} a_i = 1$ and $\sum_{i=1}^{n_0+1} a_i x_i < \delta$.
Then, lump the last two elements of $A'$ creating $A''= \{a_1,\dots,x_{a_0-1},b_{n_0}\}, b_{n_0} = a_{n_0}+a_{n_0+1}$.
Also, lump the last two elements of $X'$ creating $X''= \{x_1,\dots,x_{n_0-1},y_{n_0}\}$, $y_{n_0} = \frac{a_{n_0}x_{n_0}+a_{n_0+1}x_{n_0+1}}{a_{n_0}+a_{n_0+1}} $.
The sum of the elements of $A''$ is 1 and the weighted average of $X''$ with $A''$ is also smaller than $\delta$.
Using the induction hypotheses, for all $M>0$ large enough, there exists $B=\{ a_{i_1},\dots,a_{i_\ell}\}$ subset of $A''$ such that their sum is greater than $1-1/M$ and a subset $Z=\{ x_{i_1},\dots,x_{i_\ell}\}$ subset of $X''$ such that $x_{i_k} \leq M \delta$, for all $k=1,\dots,\ell$.
-Now, if $y_{n_0}$ is outside of $Z=\{ x_{i_1},\dots,x_{i_\ell}\}$, then we are done, because this is a subset of $X'$ as well, satisfying the result.
-But, if $y_{n_0} \in Z=\{ x_{i_1},\dots,x_{i_\ell}\}$, then at least one of $x_{n_0}$ or $x_{n_0+1}$ is smaller than $M\delta$, but not necessarily both.
This is where I'm struggling at. How can I guarantee that both of them are smaller than $M\delta$?
Thanks in advance.