Precalculus of algebra: from infinite equations to an equation with infinite series

Leave $ Gamma $, $ Delta $ Y $ X $ be infinite but enumerable sets of variables assigned a value in $[0,1]$, where $ gamma_i $, $ chi_i $ Y $ delta_i $ refer to $ i $The variable in each set.

Suppose that for every triple $ langle gamma_i, chi_i, delta_i rangle $,
$$ 1- min (1, (1- gamma_i) + (1- chi_i)) leq min (1, delta_i) $$
Does this (infinite) collection of equations imply that
$$ 1- min (1, sum ^ infty_ {i = 0} (1- gamma_i) + (1- min (1, sum ^ infty_ {i = 0} ( chi_i))) ) leq min (1, sum ^ infty_ {i = 0} ( delta_i)) $$

Some background information:

This is really for a test of solidity in the mathematical logic with which I am struggling. The test refers to a sequential calculation with infinite rules and models with formulas assigned values ​​in $[0,1]$. Intuitively, the previous inference seems valid, but I am not able to prove it. I have tried to approach the problem in a direct and opposed way. In the direct case, I get stuck with an infinite series of $ -1 $, and in the case of contrapositives, it seems that the falsity of each of the first inequalities requires a certain relationship between $ gamma_i $, $ chi_i $ Y $ delta_i $ Which I can not deduce from the falsehood of the second inequality.