what discovery rejects the idea that the atom is indestructible?

Answer 1

Simple!!

Its RADIOACTIVITY!!

Answer 2

Many discoveries prove that atoms are not indesctructible. Probably the first was radioactivity, in which atoms change from one element to another by losing bits of themselves. Nuclear fission is an extreme example of this. Nuclear fusion, as seen in hydrogen bombs and the centre of stars takes two atoms and joins them to make one bigger one, who is a way of destroying atoms and re-assembles them.