There's also a more general proof using uniform continuity; but let's do it your way.

Remember . We have

hence, using the above formula,

.

Since the curve is a bounded closed set (thus a compact set), you can choose such that the closed ball does not meet . Then we can find such that, for any in this ball, for all . Using these bounds in the previous inequality enables to bound the right-hand side by for some constant , and to conclude quickly from there.