I'm doing some exercises and came across one that has two parts, as follows:
Given a transition matrix for a Markov Chain, $\mathbf{P}$, and a vector $\mathbf{f}$, $\mathbf{f}$ is harmonic if
$ \mathbf{f} = \mathbf{P}\mathbf{f}$
$(a)$ Show that if $\mathbf{f}$ is harmonic, then
$ \mathbf{f}=\mathbf{P}^n\mathbf{f} $
for all $n$
$(b)$ Using $(a)$, show that if $\mathbf{f}$ is harmonic,
$ \mathbf{f} = \mathbf{P}^\infty \mathbf{f} $
Am I incorrect in assuming that if $(a)$ holds, then $(b)$ holds by necessity? Are there any cases where proving that something holds for all $n$ does not prove that it holds as $n$ tends to infinity?