Lets say $a_1, a_n$ are normed vectors.
Why is the set $C = \{\Sigma_{i=1}^n \lambda_ia_i: \lambda_i \ge0\}$ closed? The $\lambda$'s can be any non-negative numbers. So C is the set of all non-negative linear combinations of the n vectors.
I tried both the method with open balls, and the method with a converging sequence(show that the point the series is convergin to is in C), but still I do not see why. (PS: If it is untrue in the generalized version, you can assume that the vectors are in $\mathbb{R}^n$.)