Solution
= Solution
The proximal optimality condition gives
$$
\frac{x_k-x_{k+1}}t\in\partial f(x_{k+1}).
$$
The <subgradient inequality> therefore yields, for every $u$,
$$
\boxed{f(u)\geq f(x_{k+1})+\frac1t(x_k-x_{k+1})^T(u-x_{k+1}).}
$$
= Solution
The proximal optimality condition gives
$$
\frac{x_k-x_{k+1}}t\in\partial f(x_{k+1}).
$$
The <subgradient inequality> therefore yields, for every $u$,
$$
\boxed{f(u)\geq f(x_{k+1})+\frac1t(x_k-x_{k+1})^T(u-x_{k+1}).}
$$