Finitely generated cone ($C=\{{\sum }_{j=1}^r{\lambda }_j{a}_j:{\lambda }_j\ge 0\}$)

A finitely generated cone is a conic hull of a finite set of vectors. Every point admits a representation using linearly independent active generators, by eliminating dependence while keeping coefficients nonnegative.

A finitely generated cone is a closed convex cone. For a convergent sequence of represented vectors, use linearly independent active generators and pass to a subsequence with the same active set. Its coefficients converge through a fixed left inverse and retain nonnegativity. General linear images of closed cones need not be closed; finiteness of the generators supplies the missing argument.