Power iteration provably works if the matrix has a unique eigenvalue $\lambda$ and $\lambda>0$

Power iteration provably works if the matrix has a unique eigenvalue
$\lambda$ and $\lambda>0$

Let $A$ be a $n\times n$ real matrix and $v_0 \in \mathbb R^n$. Define a
sequence $(v_k)_k$ of $n$-dimensional real vectors by $v_k = A^kv_0 / ||
A^kv_0 ||$. Assume that $A$ has a unique eigenvalue $\lambda$ and that
$\lambda$ is real and positive. Under this assumption I would like to
prove that $v_k$ converges as $k\rightarrow\infty$.
I believe that w.l.o.g. we may assume that $A$ is in the Jordan canonical
form. Moreover, to show the statement blockwise, we may assume $A$ is a
Jordan block. Then, we could write down the components of $v_k$ explicitly
and show the convergence, although it would be quite messy.
My question is how to show the statement without getting too messy. I
would be grateful if you could give a clue (not necessarily a complete
proof).