Eigenvector of stochastic matrix

Question

$P$ is stochastic matrix, $v$ is stochastic vector, $\frac{1}{n}(v + vP + vP^{2} + \cdots + vP^{n}) \to u$. Prove that $uP = u$.

I understood that I need to show that u is eigenvector with eigenvalue 1 and understood why 1 is always eigenvalue, but can’t understand what I should do to prove what I need. Thank you for help.

Answer 1

Hint: $$\ \frac{1}{n}\left(v + vP+vP^2+ \dots+vP^n\right)P-\frac{1}{n}\left(v + vP+vP^2+ \dots+vP^n\right)\\ =\frac{1}{n}\left(vP^{n+1}-v\right)\ . $$ What does this converge to?