Proof that Markov Chains converges to the stationary distribution

Question

Let $P$ is a transition matrix of a Markov Chain, which is irreducible, aperiodic and lets assume $\pi$ is its stationary distribution: $\pi = \pi P$. Does anyone knows the proof for the following fact (given $\pi_0$ is valid probability distribution): $$\forall \pi_0 \lim_{n \to \infty}\pi_0 P^n = \pi$$ I know the proof when if $P=Q\Sigma Q^{-1}$is the eigenvector decomposition and $Q^{-1} = Q^{T}$. But how do I prove it in the general case, when the eigenvectors are not orthogonal to each other?