We know that the sequence of probability vectors $x_0, x_1, \ldots $ satisfying $x_{k+1} = P x_k$, where $P$ is a stochastic matrix will always converge to an eigenvector of eigenvalue 1 as $k \rightarrow \infty$. Is it true that the limit of any convergent sequence of probability vectors will also be a probability vector? In general is it true that $\lim_{n \rightarrow \infty} \|x_n\| = \left\| \lim_{n \rightarrow \infty} x_n \right\|$ in a normed space if this is true?
I'm having trouble thinking about this as I can't use Cauchy-Schwarz in a normed space, and didn't learn algebra as well as I should have.