Let $\mathbb{C} \subset \mathbb{R^n}$ be a convex, closed and not-bounded set. Let $d$ be a vector which $||d||=1$, then show that:
$d \in recc(\mathbb{C})$ $\iff$ $\exists$ $\{x_k\}_{k\in\mathbb{N}}\subset\mathbb{C}$ which $\lim\limits_{k \longrightarrow \infty}||x_k||=\infty$ and $\lim\limits_{k \longrightarrow \infty}(x_k/||x_k||)=d$
I am using the first definition of recession cone found here Recession cone.
I've done this implication $"\Leftarrow"$, but I can't get the another one.
If '$\Rightarrow$' would be true, since $$\lim_{k \to \infty}(x_k/||x_k||)=d$$ any $d$ would be the limit of elements with norm 1 ($||\frac{ x_k}{||x_k||}|| =1$), hence this will hold for the limit as well : $||d||=1 $.
But this is a contradiction, since in general the cone will not consist just of elements with norm $1$.
So '$\Rightarrow$' is not true.
Maybe there is something else missing ?