Asymptotically Optimal Value of a Set of Real Sequences

Let $\mathcal{C}$ be a nonempty set of sequences $(x_{N})_{N\ge1}$ of real numbers, indexed by the natural numbers $N\ge1$, and let $v$ be a real number. We say that $v$ is an asymptotically optimal value of $\mathcal{C}$ if

(i) for every sequence $(x_{N})_{N\ge1}$ in $\mathcal{C}$ and every real number $\varepsilon>0$ there is a natural number $N_{\varepsilon}$ such that $x_{N}\ge v-\varepsilon$ for every $N\ge N_{\varepsilon}$; and

(ii) some sequence in $\mathcal{C}$ converges to $v$.