TITLE: Almost convex combinations in $\mathbb R^n$
QUESTION [8 upvotes]: Working on some problems in the $C_p$-theory I discovered the following simple but amazing 
Fact. For any subset $A\subset \mathbb R^n$, non-zero vector $a\in \bar A\subset\mathbb R^n$ and $\varepsilon>0$ there are points $a_1,\dots,a_n\in A$ and real numbers $t_1,\dots,t_n$ such that $a=\sum_{i=1}^nt_ia_i$ and $$1-\varepsilon<\sum_{i=1}^n t_i\le\sum_{i=1}^n|t_i|<1+\varepsilon.$$
It seems that this fact (which can be easily proved by induction on $n$) is too elementary to be unknown. I would appreciate any reference (to a paper or textbook). Thank you.

REPLY [3 votes]: I do not know the reference, but it looks that even more may be achieved: $t_1,\dots,t_{n-1}$ may be chosen close to 0 and $t_n$ close to 1. Indeed, if the span of $a\cup A$ is spanned by linearly independent vectors $a, a_1,\dots,a_k$, $k\leqslant n-1$, then choose $a_{n} $ close to $a$,  we get $a_{n}=c a+c_1a_1+\dots+c_ka_k $, where $c$ is close to 1  and all $c_i$ close to 0, dividing by $c$ this gives a required representation for $a$.