# 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.