Summary: We study the metric projection onto the closed convex cone in a real Hilbert space \(\mathscr{H}\) generated by a sequence \(\mathcal{V} = \{v_n\}_{n=0}^\infty\). The first main result of this article provides a sufficient condition under which the closed convex cone generated by \(\mathcal{V}\) coincides with the following set: \[ \mathcal{C}[[\mathcal{V}]]: = \bigg\{\sum_{n=0}^\infty a_n v_n\Big|a_n\geq 0, \text{ the series }\sum_{n=0}^\infty a_n v_n \text{ converges in } \mathscr{H}\bigg\}. \] Then, by adapting classical results on general convex cones, we give a useful description of the metric projection onto \(\mathcal{C}[[\mathcal{V}]]\). As an application, we obtain the best approximations of many concrete functions in \(L^2([-1,1])\) by polynomials with nonnegative coefficients.