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?