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.



Almost convex combinations in $\mathbb R^n$
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 $a$. 


