This is one of those things that can of course be proven, but a proof will make it seem more complicated than the obvious reason why it's true.
You step along a path, and at each disceret step, e changes by -1, 0, or 1. You start stepping where e = rad(G). You end stepping where e = diam(G) > rad(G). You must necessarily step through every integer between them.
You could prove that by induction on the length of the path, or by contradiction with the smallest value "skipped", or by an algebraic representation of the change at each step, or other ways I'm sure.
Ex: Suppose you start with $5. You have finite number (say 8) of financial transactions. At the conclusion of each transaction, you've either gained $1, lost $1, or experienced no change. When you're done with all the transactions, you end up having $9. Necessarily, at the conclusion of at least one of those transactions, you had $6. Likewise for $7, and for $8.