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 $\|u_i-v_i\|<\frac{1}{i}.$" ?