I can see how the statement would be true if the family of functions was equicontinuous.
First, there is a point such that . In other words, for any , there exists such that implies . Since the are equicontinuous, there is such that implies for all . This gives the inequality
whenever and . This proves that is uniformly convergent to 0 on the interval .
At the moment, I am not sure if equicontinuity can be dropped.