Can someone please tell me the brief sketch (or any known reference) of the following results?


1. Why $\ell\_2$ is finitely representable in any infinite-dimensional Banach space?
2. Why every Banach space is finitely representable in $c\_0$?


A Banach space $Y$ is said to be finitely representable in some Banach space $X$ if for any finite dimensional subspace $F$ of $Y$ and $\varepsilon>0$ there exists an isomorphism $T:F\to X$ such that $\|T\|\|T^{-1}\|\leq 1+\varepsilon$.



About finite representability of Banach space
1 is very hard and 2 is very easy. For 2, take an $\varepsilon$-net $(x\_i)\_{i=1}^m$ in the sphere of the $n$ dimensional space ($m$ depends on $n$ and $\varepsilon$), and norming functionals $f\_i$'s. Now check that the map from the $n$-dimensional space into $\ell\_{\infty}^m\subset c\_0$ given by $x\to (f\_1(x), f\_2(x), \ldots, f\_m(x))$ works. 


For 1 there is a bit of easier proof if you don't care about the constant being $1+\varepsilon$ and if you are familiar with spreading models but still pretty involved. But standard reference for 1 is [Milman-Schechtman's book](https://www.springer.com/us/book/9783540167693)


