I am trying to prove the following fact :
Let $E$ be a normed vector space. Let $x_1, ..., x_n \in E$ and defined : $B = \{ \sum_{i = 1}^n \lambda_i x_i \mid (\lambda_1, ..., \lambda_n) \in \mathbb{R}_+^n \}$ Prove that $B$ is a closed set.
Here is what I 've done so far :
The set $B' = \{ \sum_{i = 1}^n \lambda_i x_i \mid \lambda_i \geq 0, \sum \lambda_i = 1\}$ is compact since the set $\{(\lambda_1, ..., \lambda_n) \mid \lambda_i \geq, \sum \lambda_i = 1 \}$ is compact. Intuitively with theset $B'$ I am calculating all points in the convex hull of $(x_1, ..., x_n)$. So the set $B$ is basically the set : $\{\lambda x \mid \lambda \geq 0, x \in B' \}$, now does it help ? I don't know. I can use the fact that if $0 \not\in B'$ then : $\{\lambda x \mid \lambda \geq 0, x \in B' \}$ is closed. Yet how to do when $0 \in B'$ ?
Thank you !