Suppose ($x_n$) is a basic sequence in a Banach space $X$, and $Y$ is a closed, infinite co-dimensional subspace of the closed span of $(x_n)$. Can we always find a subsequence ($y_n$) of ($x_n$) such that the intersection between $Y$ and the closed span of $(y_n)$ is trivial? I think the answer is 'yes' if $Y$ is finite dimensional, but I am not sure otherwise.
Edit: Indeed, it was a typo. Thanks for the bounty.