Let $M$ be a subspace of $C[0,1]$ isomorphic to $c\_0$. 


QUESTION: Is it possible to find a normalized disjoint sequence $(f\_n)$ in $C[0,1]$ such that the distance of $f\_n$ to $M$ tends to $0$ as $n$ goes to $\infty$?


Arguments in favor: 


1. If $1\leq p<\infty$, then the result is true for subspaces of $L\_p(0,1)$ isomorphic to $\ell\_p$.
2. If we consider the continuous linear projection $P$ on $C[0,1]$ onto $M$, then it was proved by N. Ghoussoub and W.B. Johnson [Math. Z. 194, 153-171 (1987); Theorem I.3] that $P$ is an isomorphism on the closed subspace $[g\_n]$ generated by some normalized disjoint sequence $(g\_n)$.


With respect to 2, if I were able to show that $[g\_n]\cap M$ is infinite dimensional or $[g\_n]+M$ is not closed, then it would be easy to get a positive answer. 



Copies of $c_0$ in $C[0,1]$ and disjoint sequences
Take sequences of clopen sets $M\_i$, $N\_i$, s.t. any two of the $N\_i$ have non empty intersection but any three have empty intersection, and the $M\_i$ are pairwise disjoint and disjoint from the $N\_i$. Let $f\_i$ be the characteristic function of $M\_i \cup N\_i$. This gives a counterexample in $C(\Delta)$, $\Delta$ a Cantor set. Doing it in $C[0,1]$ is similar but not as nice to describe. 


