Example of convex subset (unbounded) with $\text {rec} (C) = {0}$

Question

Example of convex subset (unbounded) with $\text {rec} (C) = {0}$

I've proved that for a bounded convex subset $C$ it always holds that $\text {rec} (C) = {0}$.

However, now I'm looking for an example of an unbounded convex subset with $\text {rec} (C) = {0}$.

I've tried doing some drawings, but all the time I can find some non-zero vector in $\text {rec} (C)$

Can someone help me out ?

Answer 1

Take $C=\{(a_n )_{n\in\mathbb{N}} \in \ell_{\infty} :\forall_{j\in\mathbb{N}}\hspace{0.5cm} 0\leqslant a_j \leqslant j\}.$ Then $\mbox{recc} (C)=\{0\}$ but $C$ is unbounded.