How does an inequality with respect to norms indicate linear independence?

Question

Denote $e_1,...,e_n$ an orthonormal basis of a finite-dimensional inner product space $V$, and $v_1,...,v_n$ a family in $V$ with $$\vert\vert e_i-v_i\vert\vert^2<\frac{1}{n}$$ holds for every $i\in \{1,...,n\}$. How do I show that $v_1,...,v_n$ is linearly independent, thus also a basis of V?

Answer 1

Let $\sum_{i=1}^na_iv_i=0$, where $\sum_{i=1}^n|a_i|^2=1$. Note that $|a_i|\leq 1$ for every $i$.

Thus

$1=\sum_{i=1}^n|a_i|^2=\|\sum_{i=1}^na_ie_i\|^2=\|\sum_{i=1}^na_ie_i-\sum_{i=1}^na_iv_i\|^2\leq (\sum_{i=1}^n|a_i|\|e_i-v_i\|)^2<(\sum_{i=1}^n\frac{|a_i|}{n})^2\leq 1$.

Absurd.