# What is the role of inequality in this problem? [duplicate]

Let $\{e_1,e_2,\ldots,e_n\}$ be orthonormal basis of $V$ and let $v_1,v_2,\ldots,v_n$ are vectors in $V$. If $\|e_j-v_j\|<1/ \sqrt n$ for $j=1,2,\ldots,n$ then how can we prove that $\{v_1,v_2,\ldots,v_n\}$ is basis for $V$?