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.