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.

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  • $\begingroup$ It is amazing indeed, that you can have the $t_i$ as close as you want to be all positive. If I may ask, on what parameter does the induction work? $\endgroup$ – Arnaud Mortier Jan 25 '18 at 14:18
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    $\begingroup$ So, you say that in space $\mathbb R^n$ of dimension $n$, you need $n$ terms in your sum? $\endgroup$ – Gerald Edgar Jan 25 '18 at 14:20
  • $\begingroup$ @GeraldEdgar Yes, $n$ terms. Note that $a$ is not zero! $\endgroup$ – Taras Banakh Jan 25 '18 at 18:49
  • $\begingroup$ @ArnaudMortier The induction in on the dimension $n$ and starts with the obvious case of the real line. I have made a corresponding correction to the question. $\endgroup$ – Taras Banakh Jan 25 '18 at 18:50
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    $\begingroup$ This seems to be a characterisation of the closed convex hull then, maybe up to the zero vector subtlety. $\endgroup$ – Arnaud Mortier Jan 25 '18 at 19:09

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 $a$.

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