Consider for $r,c>0$ the set
$$X_c=\{x \in \ell^1(\mathbb{N}) \mid ||x||_1=r,\,  \forall i \in \mathbb{N}: |x_i|<c\}.$$
Then I can show that $\inf_{x \in X_c} ||x||_2 = 0.$
But is it possible to compute
$$s_c:=\sup_{x \in X_c} ||x||_2 \quad ?$$
Does at least $\lim_{c \to 0} s_c=r$ hold?