  We prove a no-dimensional version of Carath\'edory's theorem: given an
$n$-element set $P\subset \Re^d$, a point $a \in \conv P$, and an integer $r\le
d$, $r \le n$, there is a subset $Q\subset P$ of $r$ elements such that the
distance between $a$ and $\conv Q$ is less than $\diam P/\sqrt {2r}$. A general
no-dimension Helly type result is also proved with colourful and fractional
consequences. Similar versions of Tverberg's theorem and some of their
extensions are also established.
