A TRACON Controller providing OJT reported their trainee failed to climb an aircraft in a timely manner and it flew below the Minimum Vectoring Altitude.

Narrative

I was working with a trainee on the ZZZ North position with all the other East Satellite functions combined as is normal for this time in the morning. My trainee is very far into an extension; and I have been trying to give him every opportunity to work situations out in order to give him the opportunity to show me he can handle this function. During the event Aircraft X departed ZZZ and climbed to 2000 ft. The trainee left the aircraft at 2000 ft. instead of climbing to 3000 ft. when he had the opportunity to do so. He eventually turned he aircraft northbound; which pointed them towards at 2500 ft. MVA. He got distracted by VFR aircraft checking in; and an aircraft requesting an IFR pickup.I should have taken over the sector earlier as he was definitely getting further and further behind; but as I stated earlier; I was trying to give him the opportunity to work the situation out and work himself out of being behind. As the aircraft was about to enter the MVA; I saw it and instructed him to climb the aircraft. I could have potentially issued a low altitude alert in this situation. However; as the aircraft was already above the obstruction; and the climb was issued; I did not. I wouldn't give the trainee so much leeway to work situations out; because I gave him so much leash. I did not notice we hadn't climbed the aircraft out of 2000 ft. until we were already right on top of the 2500 ft. MVA.