From infinitely many equations to one equation with infinite series











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



Suppose that for each triple $langlegamma_i,chi_i,delta_irangle$,
$$1-min(1,(1-gamma_i)+(1-chi_i))leqmin(1,delta_i)$$
Does this (infinite) collection of equations imply that
$$1-min(1,sum^infty_{i=0}(1-gamma_i)+(1-min(1,sum^infty_{i=0}(chi_i))))leqmin(1,sum^infty_{i=0}(delta_i))$$



Some background information:



This is actually for a soundness proof in mathematical logic I'm struggling with. The proof concerns a sequent calculus with infinitary rules and models with formulas assigned values in $[0,1]$. Intuitively the above inference seems valid but I'm not capable of proving it. I've tried to approach the problem both directly and contrapositively. In the direct case, I get stuck with an infinite series of $-1$, and in the contrapositive case, it seems that the falsity of each of the first inequalities requires a certain relationship between $gamma_i$, $chi_i$ and $delta_i$ which I'm not able to deduce from the falsity of the second inequality.










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    up vote
    2
    down vote

    favorite












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



    Suppose that for each triple $langlegamma_i,chi_i,delta_irangle$,
    $$1-min(1,(1-gamma_i)+(1-chi_i))leqmin(1,delta_i)$$
    Does this (infinite) collection of equations imply that
    $$1-min(1,sum^infty_{i=0}(1-gamma_i)+(1-min(1,sum^infty_{i=0}(chi_i))))leqmin(1,sum^infty_{i=0}(delta_i))$$



    Some background information:



    This is actually for a soundness proof in mathematical logic I'm struggling with. The proof concerns a sequent calculus with infinitary rules and models with formulas assigned values in $[0,1]$. Intuitively the above inference seems valid but I'm not capable of proving it. I've tried to approach the problem both directly and contrapositively. In the direct case, I get stuck with an infinite series of $-1$, and in the contrapositive case, it seems that the falsity of each of the first inequalities requires a certain relationship between $gamma_i$, $chi_i$ and $delta_i$ which I'm not able to deduce from the falsity of the second inequality.










    share|cite|improve this question







    New contributor




    Andy Mount is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.






















      up vote
      2
      down vote

      favorite









      up vote
      2
      down vote

      favorite











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



      Suppose that for each triple $langlegamma_i,chi_i,delta_irangle$,
      $$1-min(1,(1-gamma_i)+(1-chi_i))leqmin(1,delta_i)$$
      Does this (infinite) collection of equations imply that
      $$1-min(1,sum^infty_{i=0}(1-gamma_i)+(1-min(1,sum^infty_{i=0}(chi_i))))leqmin(1,sum^infty_{i=0}(delta_i))$$



      Some background information:



      This is actually for a soundness proof in mathematical logic I'm struggling with. The proof concerns a sequent calculus with infinitary rules and models with formulas assigned values in $[0,1]$. Intuitively the above inference seems valid but I'm not capable of proving it. I've tried to approach the problem both directly and contrapositively. In the direct case, I get stuck with an infinite series of $-1$, and in the contrapositive case, it seems that the falsity of each of the first inequalities requires a certain relationship between $gamma_i$, $chi_i$ and $delta_i$ which I'm not able to deduce from the falsity of the second inequality.










      share|cite|improve this question







      New contributor




      Andy Mount is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.











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



      Suppose that for each triple $langlegamma_i,chi_i,delta_irangle$,
      $$1-min(1,(1-gamma_i)+(1-chi_i))leqmin(1,delta_i)$$
      Does this (infinite) collection of equations imply that
      $$1-min(1,sum^infty_{i=0}(1-gamma_i)+(1-min(1,sum^infty_{i=0}(chi_i))))leqmin(1,sum^infty_{i=0}(delta_i))$$



      Some background information:



      This is actually for a soundness proof in mathematical logic I'm struggling with. The proof concerns a sequent calculus with infinitary rules and models with formulas assigned values in $[0,1]$. Intuitively the above inference seems valid but I'm not capable of proving it. I've tried to approach the problem both directly and contrapositively. In the direct case, I get stuck with an infinite series of $-1$, and in the contrapositive case, it seems that the falsity of each of the first inequalities requires a certain relationship between $gamma_i$, $chi_i$ and $delta_i$ which I'm not able to deduce from the falsity of the second inequality.







      sequences-and-series algebra-precalculus






      share|cite|improve this question







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      Andy Mount is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.











      share|cite|improve this question







      New contributor




      Andy Mount is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.









      share|cite|improve this question




      share|cite|improve this question






      New contributor




      Andy Mount is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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      asked Nov 21 at 17:16









      Andy Mount

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      New contributor




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      New contributor





      Andy Mount is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.






      Andy Mount is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.



























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