= Solution
Write $D=f(x)-m(s)$ for the predicted decrease, assuming a nonstationary point and a step at least as good as the <Cauchy point>. When $\Delta\leq\|g\|_2$, part (ii) gives $D\geq\tfrac12\|g\|_2\Delta$. Part (iv), using its weaker requested bound, then gives
$$
|1-\rho|=\frac{|f(x+s)-m(s)|}{D}\leq\frac{2\Delta}{\|g\|_2}.
$$
Consequently $\Delta\leq\tfrac12(1-\eta)\|g\|_2$ implies $\boxed{\rho\geq\eta}$.
For the radius conclusion, use the standard update rules from part (i). Suppose no subsequence of the generated gradient norms tends to zero. Then some $\varepsilon>0$ bounds those norms below eventually. If $\Delta_k\to0$, eventually $\Delta_k\leq\tfrac12(1-\eta_2)\varepsilon$, so every subsequent step is very successful by the preceding estimate. The radius is then maintained or increased, contradicting convergence to zero. Hence \b[a radius tending to zero forces a subsequence of gradient norms to tend to zero]. The conclusion uses the trust-region update rules as well as the model estimates.
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