Use properties of inner product in general inner product space to show two vectors are orthogonal

Question

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.

Answer 1

$$\left(\sum\limits_{n=1}^N(x_n,x)x_n\,\,,\,\,x-\sum\limits_{n=1}^N(x_n,x)x_n\right)=\sum_{n=1}^N(x_n,x)^2-\sum_{n=1}^N\sum_{k=1}^N(x_n,x)(x_k,x)(x_n,x_k)=$$

$$=\sum_{n=1}^N(x_n,x)^2-\sum_{n=1}^N(x_n,x)^2=0$$

since $\,(x_n,x_k)=\delta_{n,k}\,$