= Solution
Take $f$ in the <generator domain> $D(L)$. The <semigroup commutation with its unbounded generator> implies $P_tf\in D(L)$ and
$$
\frac d{dt}P_tf=P_tLf=LP_tf.
$$
Indeed, applying the bounded operator $P_t$ to the difference quotient at zero gives the right derivative $P_tLf$; <strong continuity> gives the full orbit derivative for $t>0$. Now <integration by parts> in the <Bochner integral> yields
$$
\begin{aligned}
R_\lambda Lf
&=\int_0^\infty e^{-\lambda t}P_tLf\,dt
=\int_0^\infty e^{-\lambda t}\frac d{dt}P_tf\,dt\\
&=\left[e^{-\lambda t}P_tf\right]_0^\infty
+\lambda\int_0^\infty e^{-\lambda t}P_tf\,dt
=-f+\lambda R_\lambda f.
\end{aligned}
$$
The boundary term at infinity vanishes because $\|e^{-\lambda t}P_tf\|\leq e^{-\lambda t}\|f\|$. The derivative integral exists as a <Bochner integral>, since its integrand is bounded in <norm> by $e^{-\lambda t}\|Lf\|$.
Multiplying this identity by $\lambda$ gives
$$
\lambda(\lambda R_\lambda-I)f=\lambda R_\lambda(Lf).
$$
Part (b) applies to the fixed vector $Lf\in E$, so
$$
\boxed{\lambda(\lambda R_\lambda-I)f\longrightarrow Lf
\quad\text{in }E\qquad(f\in D(L)).}
$$
The bounded operators $L_\lambda=\lambda^2R_\lambda-\lambda I$ are the <Yosida approximation> of the <semigroup generator>. The proof only applies the unbounded operator $L$ to $f\in D(L)$; the contraction and limit arguments then act on the ordinary vector $Lf$.
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