Well I know since it is $\displaystyle ||x||_{1}$ that it is absolute value of every term $\displaystyle x_{1}+x_{2}+...+x_{n}$
How about trying something like $\displaystyle \|\alpha(x_1,\cdots,x_n)\|=\|(\alpha x_1,\cdots,\alpha x_n)\|=\sum_{j=1}^{n} |\alpha x_j|=|\alpha|\sum_{j=1}^{n}|x_j|=|\alpha|\|(x_1,\c dots,x_n)\|$