Consider the following definition of closed maps, defined in the book Nonlinear Programming by Bazaraa et al.:
Let $X$ and $Y$ be nonempty closed sets in $\mathbb{R}^p$ and $\mathbb{R}^q$, respectively. Let $\mathbf{A} \colon X \to Y$ be a point-to-set map. The map $\mathbf{A}$ is said to be closed at $\mathbf{x} \in X$ if for any sequences $\{\mathbf{x}_k\}$ and $\{\mathbf{y}_k\}$ satisfying
$\mathbf{x}_k \in X, \qquad\qquad\quad \mathbf{x}_k \to \mathbf{x}$
$\mathbf{y}_k \in \mathbf{A}(\mathbf{x}_k), \qquad \mathbf{y}_k \to \mathbf{y}$
we have that $\mathbf{y} \in \mathbf{A}(\mathbf{x})$. The map $\mathbf{A}$ is said to be closed on $Z \subseteq X$ if it is closed at each point in $Z$.
I'm trying to solve the following problem (which is problem 7.7 of the aforementioned book):
Let $\mathbf{A} \colon \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n$ be the point-to-set map defined as follows. Given $\mathbf{x}, \mathbf{z} \in \mathbb{R}^n$, then $\mathbf{y} \in \mathbf{A}(\mathbf{x},\mathbf{z})$ means that $\mathbf{y} = c\mathbf{x} + (1-c)\mathbf{z}$ for some $c \in [0,1]$ and
$\|\mathbf{y}\| \le \| \lambda \mathbf{x} + (1-\lambda)\mathbf{z} \|$ for all $\lambda \in [0,1]$.
Show that the map $\mathbf{A}$ is closed for each of the following cases:
- $\|\cdot\|$ denotes the Euclidean norm; that is, $\|\mathbf{g}\| = (\sum_{i=1}^n{{g_i}^2})^{1/2}$.
- $\|\cdot\|$ denotes the $\ell_1$ norm; that is, $\|\mathbf{g}\| = \sum_{i=1}^n{|g_i|}$.
- $\|\cdot\|$ denotes the sup norm; that is, $\|\mathbf{g}\| = \max_{1 \le i \le n}|g_i|$.
I tried a lot of approaches, but I couldn't figure out what to do. Any hints and suggestions are welcome.