Proposition: Let w $\in V$ so that V is a vector space over $\mathbb{R}$ and $||w|| \neq 0$. For every v in V there exists a unique $c\in \mathbb{R}$ so that $v-cw$ is perpendicular to w.
My Proof:
Existence: Assume $w \in V$ and $||w|| \neq 0$. Let $v \in V$ be arbitrary. Seting c $\in \mathbb{R}$ to be $\frac{<v,w>}{<w,w>}$ we have $<v-cw,w>=<v-\frac{<v,w>w}{<w,w>},w>$ $=$ $<v,w>$ $-\frac{<v,w>}{<w,w>}$ $<w,w>$ $=$ $<v,w>(1-1)=0$
Uniqueness: Assume $<v-cw,w>=0$ $=$ $<v,w>-<cw,w>$ $=$ $<v,w>-c<w,w>=0$ $\implies$ $c=\frac{<v,w>}{<w,w>}$.
Is the proof correct? How do I improve it? In particular i'm concerned with the uniqueness part.
In is not particulary well written, but it is correct. In the existence proof, after the third $=$ sign I would put $\langle v,w\rangle-\langle v,w\rangle=0$; it looks more natural to me.
In the uniqueness part, I would begin with “Assume that $\langle v-cw,w\rangle=0$. Then…”