It's amazing how hidden this is on the internet, I can only find quantitive theorems which are difficult (they're great, but I want to see the original proof).
I'm looking for a simple proof of Steinitz theorem: Given a set of size $n$ with an interior point of its convex hull in $R^d$, there is a subset of size $2d$ such that the point is in the interior of the convex hull of the subset.