The greedy set is independent and meets every block

Before a nonroot block $B$ is processed, every previously chosen vertex lies in an earlier processed block: the root choice lies in the root block, and every later addition lies in the block being processed. If a vertex $x$ is newly chosen in $B$, then $x$ is different from the attachment vertex $a$. By the exposure property, $x$ has no neighbour among vertices of earlier blocks except possibly $a$. Hence it has no earlier chosen neighbour except possibly $a$.

Induct over the processed blocks, maintaining that the chosen set is independent and meets every block already processed. After the root stage, the set has one vertex, so it is independent and meets the root block. At a later block $B$, if its attachment vertex $a$ is already chosen, no vertex is added and $B$ is met at $a$. Otherwise a vertex $x$ of $B$ other than $a$ is added. By step 1.1, its only possible earlier chosen neighbour is $a$, which is not chosen; hence adding $x$ preserves independence, and $x$ meets $B$. No vertices are removed, so all earlier blocks remain met.

The finite induction therefore gives both properties after all blocks have been processed. The resulting set is the output $I$, which is independent and meets every block of $H$.