$\mathbf{Theorem}$: Suppose $X$ is an irreducible Markov chain with transition matrix $P$. Let $\lambda$ be an invariant measure for $P$, i.e. $\lambda P = \lambda$. Suppose that some $\lambda_k > 0$. Then all $\lambda_i > 0$.
$\mathbf{Proof}$: We have $\lambda = \lambda P = \lambda P^n$ for all $n$. Let $p^{(n)}_{ij}$ be the entries of $P^n$. We know by irreducibility that there is some $n$ such that $p_{ki}^{(n)} > 0$. As the chain is irreducible we have $\lambda_i = \sum\limits_{j} \lambda_j p^{(n)}_{ji} \geq \lambda_kp_{ki}^{(n)} > 0$ so we are done.
I don't understand where the second last inequality comes from (where we replace the sum with a single term). It should hold only if the terms we are deleting are all non-negative and as $p_{ji}^{(n)}$ is always non-negative that would require that $\lambda_j$ is non-negative for all $j$ which is (almost) what we set out to prove.
Can anyone tell me what I am missing here?
A measure is non-negative by definition. A measure that is instead allowed to take negative values is called a signed measure. The point of the theorem is that each state has a strictly positive probability.