Proof with stationary distribution

Question

Let $\pi(k)$ the stationary distribution of the Markov Chain. Show that if $$p_{ij}^{(n)}\geq\varepsilon$$ for some $i,j,n,\varepsilon$ then $$\pi(j)\geq \varepsilon \pi(i)$$

I'm litle lost here $$\pi(j)=\sum_i p(i,j)\pi(i)$$ $$p_{ij}^{(n)}\rightarrow \pi(j)$$ thus $$p_{ij}^{(n)}\rightarrow \pi(j)\Rightarrow \pi(j)\geq\varepsilon$$

but do not know how to proceed and check the relationship

Answer 1

Hint: Show that $\pi(j)=\sum_i p^{(2)}(i,j)\pi(i)$. And then the same with 2 replaced by 3, etc.