# Proving The Fundamental Theorem of Markov Chains

I am reading these university notes (https://web.stanford.edu/class/cs265/Lectures/Lecture14/l14.pdf) on Page 3.

The following statement is written:

Theorem 1 (The Fundamental Theorem of Markov Chains): Let $$X_0, X_1, \ldots$$ be a Markov chain over a finite state space, with transition matrix $$P$$. Suppose that the chain is irreducible and aperiodic. Then the following hold:

1. There exists a unique stationary distribution, $$\pi = (\pi_1, \pi_2, \ldots)$$ over the states such that: for any states $$i$$ and $$j$$, $$\lim_{{t \to \infty}} Pr[X_t = i|X_0 = j] = \pi_i.$$
1. For each state $$i$$, $$\pi_i = \frac{1}{E[\min(t:X_t=i)|X_0=i]}$$, namely $$\pi_i$$ is the inverse of the expected return time of state $$i$$.
1. $$\pi$$ is a left eigenvector of matrix $$P$$, with eigenvalue 1, namely the vector-matrix product $$\pi P = \pi$$.

My Question: I am trying to find a proof for 2:

For each state $$i$$, $$\pi_i = \frac{1}{E[\min(t:X_t=i)|X_0=i]}$$, namely $$\pi_i$$ is the inverse of the expected return time of state $$i$$.

I found some corresponding videos (i.e. same author as the notes, Dr. Wootters) to these notes in which a proof for 1 is provided (https://www.youtube.com/watch?v=U3nJFDs7fUc&list=PLkvhuSoxwjI_JL7GYcJHK7-EK55t0KYGO&index=30), but I have not been able to find a proof for 2.