Convex cone necessary and sufficient condition

Question

Let C be a nonempty set in $R^n$ . Show that C is a convex cone if and only if $x_1, x_2 \in C $ implies that $\lambda_1\cdot x_1 + \lambda_2\cdot x_2 \in C $ for all $\lambda_1 ,\lambda_2 \geq0$.

Answer 1

If $C$ is a convex cone use that $\lambda_i = \frac{\lambda_i} {\lambda_1 + \lambda_2} \cdot (\lambda_1 + \lambda_2)$, for $i=1,2$.