We set $r_\infty := \lim r_k$ and
$$h_k := (I + P_0) \circ (I + P_1) \circ \cdots \circ (I + P_{k-1}).$$
Explain why $r_\infty$ is well defined, and show that $\hat{h}_k(r_k) \leqslant r_0$ for all $k \geqslant 1$. Deduce that the series $h$ of question E satisfies $\hat{h}(r_\infty) \leqslant r_0$, thus that $\rho(h) \geqslant r_\infty$.