The goal is to prove that every basis in a finite-dimensional Hilbert space is a Riesz basis, i.e., there exist constants $A>0$ and $B>0$ for the basis $\{x_k\}$ such that: $$ A \sum_n |a[n]|^2 \leq \left\|\sum_n a[n] x_n \right\|^2 \leq B \sum_n |a[n]|^2 $$ for all $\{a[n]\} \in \ell_2$.
There is an argument given in Yonina Eldar's book for the right inequality which I easily understand, for the left inequality the argument is given that no non-trivial sequence $a[n]$ satisfies $\left\|\sum_n a[n] x_n \right\|^2=0$. This point I fail to understand, as I don't see how the lower bound $A$ is guaranteed to be strictly greater than $0$. Is there some defect in the argument?
If you consider only finite dimensional Hilbert spaces, the "sequences" are in fact just vectors of some given length, i.e., the dimension of the space. Thus, the set of all those vectors $\{a[n]\}$ with $$ \sum_n |a[n]|^2 =1 $$ is compact and the continuous function $$ \{ a[n] \} \mapsto \left\lVert \sum_n a[n] x_n \right\rVert^2 $$ attains its minimum, which can't be zero.