I have some points $x_1,\dots,x_n$ in $\mathbb{R}^j$. I define the set $C(x_1,\dots,x_n)= \left\{\sum_{i=1}^{n} \lambda_i x_i : \lambda_1+\dots+ \lambda_n = 1, \lambda_i \ge 0 \right\}$. The closure of a set is given by all its cluster points. We call a set $S$ closed if $\overline{S}=S$. I now want to prove constructively, meaning without the use of the law of excluded middle, that the set $C(x_1,\dots,x_n)$ is closed. To this end let $a^{(m)} = (a_1^{(m)},\dots, a_n^{(m)})$ be a converging sequence such that $\sum_{i=1}^{n} a_i^{(m)} = 1$ and $a_i^{(m)} \ge 0$. Obviously we have for the limit $a = (a_1,\dots,a_n)$ of this sequence that $\sum_{i=1}^{n} a_i =1$ and $a_i \ge 0$. Since any converging sequence in $C(x_1,\dots,x_n)$ is of the form $a_1^{(m)}x_1 + \dots+ a_n^{(n)}x_n$ and its limit $a_1 x_1 + \dots + a_n x_n$ is still in $C(x_1,\dots,x_n)$, we get that the set is closed.
However, I have be told that constructively this set is NOT closed. But I don't see any flaws in my prove so I am a little confused.