Vtyjkyuk

I have a Hilbert space H. I fix elements $x_1,dots,x_n$ in $H$ and define the convex hull $C(x_1,dots,x_n) = sum_i=1^n a_i x_i : a_i ge 0 text and a_1 + dots + a_n = 1$. The relative interior of the convex hull is given by $mathringC(x_1,dots,x_n) = sum_i=1^n a_i x_i : a_i > 0 text and a_1 + dots + a_n = 1$. We have given that $0 notin mathringC(x_1,dots,x_n)$. A proof now wants to show that for any $k in mathbbN_+$
beginalign*
d(-frac1k sum_i=1^n x_i , C(x_1,dots,x_n)) >0.
endalign*
(I know that this is trivial, but I am trying to understand the proof given in the notes.)
They suppose $d(-frac1k sum_i=1^n x_i , C(x_1,dots,x_n)) =0$, hence there exists $ a in mathbbR^n$ with $a_i ge 0$ and $a_1+dots+a_n = 1$ such that beginalign*
-frac1k sum_i=1^n x_i = sum_i=1^n a_i x_i.
endalign*
By this they conclude $0 in mathringC(x_1,dots,x_n)$, a contradiction.
I struggle with the final conclusion. From the equality we get
beginalign*
0 = sum_i=1^n (a_i + frac1k) x_i
endalign*,
but we don't have $a_1 + dots + a_n + fracnk = 1$.

We know that $- sum_i=1^nx_i / n in C(x_1,dots,x_n)$ and $sum_i=1^nx_i / n in C(x_1,dots,x_n)$. Hence also $ 0 = - sum_i=1^nx_i /(2 n) + sum_i=1^nx_i /(2 n) in C(x_1,dots,x_n)$ and in particular in the relative interior by definition.