 Solution

The asymptotic equipartition property is $-n^{-1}{\log }_{2}p(X_1,\dots ,X_n)\to H$ in probability. For an i.i.d. source this is the weak law applied to $-{\log }_{2}p(X_i)$, and independence gives $H(X_1,\dots ,X_n)=nH(X)$. Thus its information rate and AEP constant are both $H(X)$.