Prove that if $e\_1, \dots e\_n$ is an orthonormal basis of $ V$ and $v\_1, \dots , v\_n$ are vectors in $ V$ such that
$$\|e\_j - v\_j\| < \frac{1}{\sqrt{n}}$$ for each j, then $v\_1, \dots v\_n$ is a basis of $V$.

I have fiddled around with this for a while now unfortunately nothing came close, so I'll just say I have no idea how to do this. Help :(?