Let $X$ be a Banach space with a Schauder basis, and $Y$ a Banach space. Let $P\_N$ denote the coordinate projections relative to the basis of $X$, and let $X\_N$ denote their ranges. Specifically, $\{x\_n\}\_{n=1}^{\infty}$ is a basis in $X$ and 
$$P\_N(x)=\sum\_{n=1}^Na\_nx\_n,\quad\hbox{with}\quad x=\sum\_{n=1}^{\infty}a\_nx\_n$$
Assume that for each $N$, the finite dimensional space $X\_N$ is isometric to a subspace of $Y$. Does it follow that $X$ is isometric to a subspace of $Y$? 


If $Y=L\_p$ with $1\leq p\leq 2$ then the answer is affirmative, but the general case escapes me.



If the ranges of all f.d basis projections in $X$ are isometric to subspaces of $Y$, does it follow that $X$ is isometric to a subspace of $Y$?
It is false for $L\_p$. Consider the Haar basis. The span of the first $n$ basis vectors is isometrically isomorphic $\ell\_p^n$, but $L\_p$ does not embed even isomorphically into $\ell\_p$ if $p\not=2$. 


