A trouble about the Simons’ inequality

Question

I have a trouble in the proof to Simons’ inequality:

About prove that:

$\displaystyle \inf_{x \,\in\, C_1} \sup_{B} (x) \le \sup_{B} (\lim_{n} \sup (x_n)) \Longrightarrow \sup_{B} (\lim_{n} \sup (x_n)) \ge \inf \{\sup_{B} (x) \,; x \,\in\, \text{conv}\{x_n\}\} $

$\text{conv}$: convex hull

Any hints would be appreciated.

Answer 1

Note that:

$\displaystyle \text{conv}\{x_n\}\subset C_1\subset \overline{\text{conv}}\{x_n\}\Longrightarrow \inf \{\sup_{B} (x) \,; x \,\in\, \text{conv}\{x_n\}\}=\inf_{x \,\in\, C_1} \sup_{B} (x)$