Sequences in $L_{p}(1Let $1<p<\infty$. Johnson and Schechtman (Multiplication operators on $L(L_{p})$ and $l_{p}$-strictly singular operators, 2008, DOI: 10.4171/JEMS/141, eudml, arxiv) observed that if $(x_{n})_{n}$ is a sequence in $L_{p}$ that is equivalent to the unit vector basis for either $l_{p}$ or $l_{2}$, then $(x_{n})_{n}$ has a subsequence $(z_{n})_{n}$ which spans a complemented subspace of $L_{p}$. Is there a stronger result? More precisely,
Question: Does the sequence $(x_{n})_{n}$ have, for every $\epsilon>0$, a subsequence $(z_{n})_{n}$ which spans a $(1+\epsilon)C_{p}$-complemented subspace of $L_{p}$? where the constant $C_{p}$ depends only on $p$.
Thank you!
 A: The answer is no.  Let $2<p<\infty$ and in $\ell_p \oplus_p \ell_2$ (which embeds isometrically into $L_p$) consider $x_n := e_n \oplus \alpha \delta_n$, where $(e_n)$; respectively, $(\delta_n)$, is the unit vector basis of $\ell_p$; respectively, $\ell_2$, and $0<\alpha <1$. One can show that the norm of any projection onto the closed span of any subsequence of $(x_n)$ is at least $M(\alpha,p)$, where $M(\alpha,p)\to \infty$ as $\alpha \downarrow 0$.  Maybe this is pointed out in Rosenthal's paper on the span of independent random variables in $L_p$.  It follows also from more general considerations in the complemented subspaces problem paper of Lindenstrauss and Tzafriri.
You can build the same sort of example on the other side of $2$, reversing the roles of $p$ and $2$.  
When $(x_n)$ is equivalent to the unit vector basis of $\ell_p$ and $2<p<\infty$, given any $\epsilon >0$ there is a subsequence of $(x_n)$ that spans a $1+\epsilon$-complemented subspace.  This is in Kadec-Pelczynski.  
Something that I do not know at this moment and maybe never knew is whether a sequence in $L_p$, $1<p< 2$, that is equivalent to the unit vector basis of $\ell_2$ must have a subsequence that spans a $C_p$-complemented subspace of $L_p$.  
