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Proof with convex sets

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A Theorem states: Convex Sets are closed under convex combinations. That is, if C is a convex set, and if x1,x2,...,xm belongs to C then for all non-negative real numbers %1,%2,...,%m are non-negative real numbers such that $1+$2+....+$m=1, we have
$1x1 + $2x2 + ... + $mxm belonging to C.

Furthermore, I want to prove that if x1,x2,x3,x4 are in the convex set C, and if $1,$2,$3,%4 are non-negative real numbers such that $1+$2+$3+$4=1 then $1x1+$2x2+$3x3+$4x4 belonging to C
(please give full proof)

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Solution Summary

This is a proof regarding convex sets and non-negative numbers

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