Consider a Markov chain with the states 0,1,2,3,4,5,6 and transition matrix $$ P=\begin{pmatrix}\frac{1}{5} & \frac{3}{5}& 0 & 0 & \frac{1}{5} & 0 & 0\\0 & 0 &1 & 0 &0&0&0\\0&\frac{1}{3}&0&\frac{2} {3}&0&0&0\\0&1&0&0&0&5&0\\0&0&0&0&0&1&0\\0&0&0&0&0&0&1\\0&0&0&0&1&0&0\end{pmatrix} $$
Show that $$ (p_{04}^{(n)})_{n\in\mathbb{N}} $$ does not converge as $ n\to\infty$. Determine limsup and liminf of the sequence.
I am wondering why the sequence does not converge because as far as I see it is $$ p_{04}^{(n)}=\left(\frac{1}{5}\right)^n. $$
which goes to 0 as $ n\to\infty$.
Hint: here the notation is:
$$ p^{(n)} = n\text{th power of }p\\ p^{(n)}_{04} = (n\text{th power of }p)_{04} $$