$($Hierarchy Theorems$)$ You may assume without saying so that any reasonable$-$looking function

$($logarithms$,$ polynomials, exponentials, and combinations thereof$)$ is time$-$constructible.

The time complexity class $P=u_k=1^{}$TIME$(n^k)$ consists of all languages decidable in polynomial

time. The time complexity class EXP $=u_k=1^{}$TIME$(2^{n^k})$ consists of all languages decidable in

exponential time $($i$.$e$.,$ in time that is an exponential of a polynomial$).$

$($a$)$ Show that PsubeTIME$(n^{logn}).$

$($b$)$ Use the time hierarchy theorem to show that EXP $TIME(n^{logn}).$

$($c$)$ Combine parts $($a$)$ and $($b$)$ to conclude that $P$ExP.