Can a child have too many Bananas? Like, is it bad for them?

Lily keeps eating heaps because really she doesn't eat much else other than apples and bananas and the occassional cheese or jam sandwich. Other than that it is hard to get food into her as she is just so fussy. She will usualy only have 2 meals a day tops with maybe 1 snack but thats pushing it. But she will eat all the bananas i can possible give her but just wanted to know if it'll do her any harm?

Any ideas on how to get her to eat more or is this just the way she will be? She will choose fruit over junk any day which is absolutely great IMO but she doesnt eat much as i said and i'm worried she will become underweight or malnurished(sp?). She is a skinny girl as it is, i dont want to make it any worse but she's a stubborn one!