I'm reading Kenneth Falconer's "Fractal Geometry" and it mentions on page 35 that box-counting dimension is not finitely stable. That is, in general it is not always the case that
$$ \dim_B (E \cup F) = \max (\dim_B E, \dim_B F)$$
But rather, this property is only true for the upper box-counting dimension (box-counting dimension defined by taking the upper limit).
I've been trying to think of a counter example to see why this is, but I can't think of one. Is there any simple example? Or does it require complicated constructions to show it's not true in general.
