Let $\{x_n\}_{n=1}^N$ be an orthonormal set in an inner product space $V$ with inner product $(\cdot,\cdot)$.
I am trying to show that $\left(\sum\limits_{n=1}^N(x_n,x)x_n,x-\sum\limits_{n=1}^N(x_n,x)x_n\right)=0.$
Reed & Simon, Functional Analysis, Theorem II.1 mention this is a “short computation using the properties of inner products” but I am having trouble with this.