Prove the version of Carath ́eodory’s theorem for cones that is: Given any vector space E of dimension m, for any (non void) family S=(vi)i∈L of vectors in E, the conecone (S) spanned by is equal to...

Prove the version of Carath ́eodory’s theorem for cones that is: Given any vector space E of dimension m, for any (non void) family S=(vi)i∈L of vectors in E, the conecone (S) spanned by is equal to the set of positive combinations off a milies of m vectors in S.