How to prove 2-EXP != EXP
It is a direct consequence of the time hierarchy theorem:

Time Hierarchy Theorem: if f and g are time-constructible functions and $f(n) \log f(n) =
o(g(n))$ (e.g. $f(n) = n^2, g(n) = n^3$), then $DTIME(f(n)) \subsetneq DTIME(g(n))$