$\"\"$

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.