I want to show that if $\{e_\alpha\}$ is an orthonormal basis for the Hilbert space $H$, then the set $\{\alpha: \langle v, e_\alpha\rangle > 1\}$ is finite for any $v \in H$. This is immediately true by Bessel's inequality, but I was told to not use Bessel's inequality or any of its corollaries. With this condition, I'm not sure where to begin.
I've thought about considering the functional $f(x) = \langle v, x \rangle$ and maybe showing its operator norm is infinite, but I can't see where to go from here.
Here goes... assume that $\{\alpha : \langle v,e_\alpha\rangle =1\}$ is infinite and extract a sequence $(\alpha_n)_n$ of such indices $\alpha$.
Consider the vector $\sum_{n=1}^\infty \frac1n e_{\alpha_n} \in H$, it is easy to show that this series converges since the sequence of its partial sums is Cauchy. But we then have $$\left\langle v,\sum_{n=1}^\infty \frac1n e_{\alpha_n}\right\rangle = \sum_{n=1}^\infty \frac1n \underbrace{\langle v,e_{\alpha_n}\rangle}_{> 1} > \sum_{n=1}^\infty \frac1n = +\infty$$ which is a contradiction.