I'm doing some grammatical analysis for uni, and I have come across this utterance which has thrown me off a little bit: "The lady gave them something to eat". So far I have analysed:

At the clause level:
The lady = Subject
gave = Verb
them = Indirect Object
something to eat = Direct Object

And at the phrase level:
The = determiner
lady = noun
gave = main verb
them = personal pronoun
something = indefinate pronoun
to = ?
eat = ?

My problem is the 'to eat' part. I think the 'to' is a prepostion, but the presence of the verb makes me think it might be the particle. Can you just have a verb sitting on its own like that?