I have tried to prove this using the Cauchy criterion for HK-integration but I have been unsuccessful thus far. I have various fancy theorems at my disposal such the MCT and the DCT but I can not think of a way to prove this seemingly basic result. If someone knows a way to prove this please let me know. I don't need an entire proof just something to get me started and I can fill out the details.
Perhaps my claim is wrong, in which case a counter-example is sufficient.