Implementing an interval tree using arrays?
To a mathematician "prove or disprove" is a very strict requirement. I do not know how serious I should take this.

Every tree that has been implemented using pointers also has a array implementation. The trick is to do the addresses yourself as indexes to an array. That is usually called a cursor implementation.

Binary trees can be implemented, as you say, storing children of node $i$ at indices $2i$ and $2i+1$. This is sometimes called an implicit representation as no explicit structure is stored. The method is  alwaus possible, but only advisable when the tree is perfect (all levels completely filled) or near perfect, like the complete trees used to store a binary heap. Otherwise it is a waste of space. When some node $i$ is missing in the tree then also nodes $2i$ and $2i+1$ are missing, and $4i,\dots, 4i+4$, etc.