= Solution
{c}
Put $z=x-\alpha\nabla f(x)$ and $\tau=\alpha\lambda$. The <proximal operator> for the <L1 norm> separates into independent scalar <minimization problems>:
$$
p_i=\mathop{\arg\min}_{u\in\mathbb R}\left\{\tau|u|+\frac12(u-z_i)^2\right\}.
$$
The <subgradient optimality condition> is $0\in p_i-z_i+\tau\partial|p_i|$. By the <subgradient of the absolute value>, a positive solution must satisfy $p_i=z_i-\tau>0$, and a negative solution must satisfy $p_i=z_i+\tau<0$. At zero, the <subgradient optimality condition> holds exactly when $|z_i|\leq\tau$. The squared term makes the <objective function> a <strictly convex function>, so these cases give its unique <minimiser>.
Thus the <proximal operator> is the <soft-thresholding operator>
$$
S_\tau(t)=
\begin{cases}
t-\tau,&t>\tau,\\
0,&|t|\leq\tau,\\
t+\tau,&t<-\tau,
\end{cases}
\qquad p_i=S_\tau(z_i).
$$
Consequently the requested <proximal gradient mapping>, rather than just the <proximal operator>, is
$$
\boxed{[G_\alpha(x)]_i=\frac{x_i-S_{\alpha\lambda}(x_i-\alpha\partial_i f(x))}{\alpha},\qquad i=1,\ldots,n.}
$$
The corresponding <proximal gradient method> update is $x_i^+=S_{\alpha\lambda}(x_i-\alpha\partial_i f(x))$. \b[Small <coordinates> are set exactly to zero.] When $\lambda=0$, the <soft-thresholding operator> is the identity and the <proximal gradient mapping> reduces to $\nabla f(x)$.
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