Let $V$ be an inner product space and $e_1, ... , e_r$ be an orthonormal list. Suppose $v \in V$ such that $\lVert v - e_r \rVert < 1$.
Prove that $e_1, ... ,e_{r-1}, v$ is linearly independent.
I was thinking of showing that $v$ and $e_r$ are dependent so $e_1, ..., e_{r-1}, v$ is independent. My attempt uses properties of the norm and inner product, but I am stuck so I think I need to approach this differently. I also think that this might need the usage of the Triangle Inequality Theorem or equality in the Cauchy-Schwarz Theorem. Any help on this would be great!
My attempt: $$ \lVert v - e_r \rVert < 1 \iff \langle v - e_r, v - e_r \rangle < 1 \\ \iff \langle v, v - e_r \rangle - \langle e_r, v - e_r \rangle < 1 \\ \iff \overline{\langle v - e_r,v \rangle} - \overline{\langle v - e_r,e_r \rangle} < 1 \\ \iff \overline{\langle v, v \rangle - \langle e_r, v \rangle} - \overline{\langle v, v \rangle - \langle e_r, e_r \rangle} < 1 \\ \iff \langle v, v \rangle + \langle e_r, v \rangle - \langle v, v \rangle + \langle e_r, e_r \rangle < 1 \\ \iff \langle e_r, v \rangle + 1 < 1 $$
This is where I got stuck.