= Solution
Irreducibility and the presence of another state imply $P_{ii}<1$: otherwise $i$ would be absorbing and could not reach that other state. Hence the proposed matrix has nonnegative entries and row sums $\sum_{j\ne i}P_{ij}/(1-P_{ii})=1$.
Observe the original <Markov chain> only at times when it changes state. Starting from $i$, the next different state is $j\ne i$ with <probability>
$$
\sum_{r=0}^\infty P_{ii}^rP_{ij}=\frac{P_{ij}}{1-P_{ii}}.
$$
Each holding run has a finite geometric length almost surely. The <Strong Markov property> at the successive change times therefore identifies the observed process as the <discrete-time jump chain> with <transition matrix> $\widetilde P$.
Any positive-<probability> path of the original chain gives a path of this new chain after repeated consecutive states are deleted. Every remaining edge has positive $\widetilde P$ <probability>, so \b[$\widetilde P$ is irreducible]. For recurrence, starting at any state $i$, repeated application of the Markov property at return times shows that the original recurrent chain visits $i$ infinitely often almost surely. A single sojourn cannot account for infinitely many visits, because every sojourn is finite almost surely. There must therefore be infinitely many separate visits to $i$ in the observed chain too. Thus \b[$\widetilde P$ is recurrent], proving that <recurrence survives deletion of self-transitions>.
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