Consider and are the position functions of the two runners.

The difference in position function of the two runners is .

Consider be the starting time and be the ending time.

Two runners start a race at the same time, .

Two runners finish a race at the same time, .

The position functions and are continuous and differentiable.

Apply mean value theorem,

and .

After some time,

Substitute and in the above expression.

After some time, the two runners have the same speed during the race.

After some time, the two runners have the same speed during the race.