Suppose $x_1, x_2, \ldots, x_n$ are linearly independent elements of a normed linear space $X$. Show that there is a constant $c>0$ with the property that for every choice of scalars $\alpha_1, \ldots, \alpha_n$ we have $\|\alpha_1x_1+\cdots+\alpha_nx_n\|\geq c(|\alpha_1|+\cdots+|\alpha_n|)$
I tried doing this by contradiction but I am stuck.