In the paper “Commutators on $ \ell_\infty$ ” by Dosev and Johnson, in Lemma 4.2 Cas II, the authors have said that “There exists a normalized bock basis $ \{u_i\}$ of $ \{x_i\}$ and a normalized block basis $ \{v_i\}$ of $ \{y_i\}$ such that $ \|u_i-v_i\|<\frac{1}{i}.$ Does anyone have any idea to prove this?

More elaborately, we have two subspaces $ X$ and $ Y$ of $ \ell_\infty$ both isomorphic to $ c_0.$ $ \{x_i\}$ and $ \{y_i\}$ are bases of $ X$ and $ Y$ respectively which are equivalent to standard base of $ c_0$ . We have also $ X\cap Y=\{0\}$ and $ d(X,Y):=\inf\{\|x-y\|:x\in X,y\in Y, \|x\|=1\}=0.$ Now how to show “There exists a normalized bock basis $ \{u_i\}$ of $ \{x_i\}$ and a normalized block basis $ \{v_i\}$ of $ \{y_i\}$ such that $ \|u_i-v_i\|<\frac{1}{i}.$ ” ?