Does the convergence of a matrix iteration imply convergence of the matrix entries?

Question

Let's say I know that $$ \lim_{k\to\infty}\mathbf{A}^k\mathbf{x=x}^*\,. $$ Does it then follow that each of the entries $a_{i,j}^{(k)}$of $\mathbf{A}^k$ converge to a limit?