If you have a constant velocity over some time interval, how can you say that the average velocity is equal to that constant velocity?
Solution 1:
An average is always between the minimum and maximum. So, the average velocity is between the minimum velocity and the maximum velocity. Since the velocity is constant, the minimum and maximum velocities equal this constant, so the average velocity must be that constant.
In symbols, $$c = v_{\text{min}} \le v_{\text{average}} \le v_{\text{max}} = c,$$ which implies that $v_{\text{average}} = c$.