Lime is generally used for eventual deterioration of tree stumps. Takes more than one year for full removal. Will have to be reapplied on a regular basis. Not a smart thing to do in such close vicinity of children playing area.
Another way is to build a few fires on the stump. Has to be really hot to get the moisture out, and usually takes more than one fire. Long-burning fuel, like charcoal and hardwood (oak for instance) is needed.
Digging around to expose the major roots at the stump, then using a chainsaw to cut these roots can help in either case. Whatever method has its own type risks and precautions.