Prove that the set $S=\{(x,y)\in R^2\,|\,0\leq x\leq 1,y\in R \}$ has no extreme points

Consider the subset of the plane, $S = \{ (x,y) \mid 0\le x \le 1 \}$, a closed vertical strip of width $1$.

The negation of $(x,y)$ being an extreme point of $S$ is that there do exist two points in $S$ such that $(x,y)$ is their midpoint.

An easy way to show this (since the strip extends infinitely up and down) is to adjust the vertical coordinate plus and minus by an equal amount. That is, taking any $(x,y) \in S$, we see:

$$ (x,y) = \frac{1}{2} ((x,y+1) + (x,y-1)) $$

So no point $(x,y) \in S$ is an extreme point of $S$.