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 book][1] 


  [1]: http://1%20is%20very%20hard%20and%202%20is%20very%20easy.%20For%202,%20take%20an%20$%5Cvarepsilon$-net%20$(x_i)_%7Bi=1%7D%5Em$%20in%20the%20sphere%20of%20the%20$n$%20dimensional%20space%20($m$%20depends%20on%20$n$%20and%20$%5Cvarepsilon$),%20and%20norming%20functionals%20$f_i$'s.%20Now%20check%20that%20the%20map%20from%20the%20$n$-dimensional%20space%20into%20$%5Cell_%7B%5Cinfty%7D%5Em%5Csubset%20c_0$%20given%20by%20$x%5Cto%20(f_1(x),%20f_2(x),%20%5Cldots,%20f_m(x))$%20works.%20For%201%20there%20is%20a%20bit%20of%20easier%20proof%20if%20you%20don't%20care%20about%20the%20constant%20being%20$1+%5Cvarepsilon$%20and%20if%20%20you%20are%20familiar%20with%20spreading%20models%20but%20still%20pretty%20involved.%20Standard%20reference%20for%201%20is%20Milman-Schechtman%20book%20https://www.springer.com/us/book/9783540167693