About an argument in the paper “Commutators on $\ell_\infty$” by Dosev and Johnson

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 $$\|u_i-v_i\|<\frac{1}{i}.$$” ?