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 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 $\|T\|\|T^{-1}\|\leq 1+\varepsilon$.

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 $\|T\|\|T^{-1}\|\leq 1+\varepsilon$.