This is a follow-up to the question I posted earlier this week.
Consider, for a fixed sequence $(a_n)_n\in\ell^2$ the subspace $C=\{(x_n)_{n}\in\ell^2 : |x_n|\le a_n\text{ for all }n\in\mathbb{N}\}\subset\ell^2.$ Is this set convex in $\ell^2$?
According to the book [An introduction to Nonlinear Analysis, by Martin Schechter, page 175] it is true, but there is no proof. Can someone help me out?