I'm in an argument with someone who claims that a two mile in 7:59 does not imply that one mile (at some point within the two miles) was covered in under 4:00.  This is obviously wrong, but I'm not sure how to create a proof showing otherwise.

Does anyone have an idea for how to go about this?

If $x \geq 4$ and $y \geq 4$ then $x+y \geq 8.$ 

EDIT: on André's extra credit problem, use Beni's way of writing, time function $f,$ then define $g(m) = f(m+1) - f(m)$ with $0 \leq m \leq 1.$ We know $f(0) = 0, \; f(2) = 8.$ So, $g(0) + g(1) = 8.$ If both $g(0), g(1)$ are $4,$ we are done with André's problem. If one of the pair is above 4, the other is below 4. So, by the Intermediate Value Theorem, there is then some other $0 < m < 1$ such that $g(m) = 4.$ 
