Proof about sequences and $\epsilon-$ inequalities in a compact set $X$

Question

Could someone please explain why in the proof

1. $\Vert y-\overline x\Vert\ge\epsilon>0\ ?$

and

2. $f(x_k)+w_k^t\beta(x_k)\le f(\overline x)+w_k^t\beta(\overline x)\ ?$

In the theorem, $X(w)=\{y:y\ minimize \ f(x)+w^t\beta(x) \text{ over } x\in X\}$

Theorem.

Thanks in advance for your help

Answer 1

1. As $x_k\to y$ and $\|x_k-\bar x\|\ge \epsilon$ for all $k$ (in $\mathscr K'$), we cannot have $\|y-\bar x\|<\epsilon$.

2. By assumption, $x_k\in X(w_k)$.