I have to show that
$\frac {1}{N}H(X_1,...,X_N)\le H(X_1)$.
for a stationary stochastic process.
I know that $H(X_1,...,X_N)=\Sigma _{i=1}^N H(X_i|X_1,...,X_{i-1})$.
So far I have plugged that into the original inequality to find that
$\frac {1}{N}\Sigma _{i=1}^N H(X_i|X_1,...,X_{i-1})=\frac {1}{N}(\Sigma _{i=1}^{N-1} H(X_i|X_1,...,X_{i-1})+H(X_N|X_1,...,X_{N-1}))\le \frac {1}{N}(\Sigma _{i=1}^{N-1} H(X_i|X_1,...,X_{i-1})+H(X_1)$
I have also previously shown that $H(X_N|X_1,...,X_{N-1}))\le H(X_1)$
Should I proceed by induction or can anybody recommended a different way?