Yes, $P^n$ converges to the matrix Q with $Q_{i,j}=0$ if $j\ne\text{null}$ and $Q_{i,j}=1$ if $j=\text{null}$. This is because $P^n_{i,j}$ is the probability that if started in $i$ after $n$ moves we are in state $j$. Take $\lambda>0$ such that $P_{i,\text{null}}\ge\lambda$ for all $i$. Then $P^n_{i,j}\le (1-\lambda)^n\longrightarrow 0$ for all $j\ne\text{null}$. 

(Note that the question is not about almost sure behavior but about a limit of matrices, so Borel-Cantelli is not relevant to the question as stated.)