I am reading the book Naive Lie Theory
It proves that any isometry of $R^n$ that fixed the origin O is the product of at most n reflections in hyperplanes through O.
The proof is elementary and by induction. However, I cannot understand the arguments.
'Now suppose that $f$ is an isometry that fixes O and the result is true for $n=k-1$. If $f$ is not the identity, suppose $v \in R^k$ is such that $f(v)=w \neq v$.
Then the reflection $r_u$ in the hyperplane orthogonal to $u=v-w$ maps the subspace $Ru$ of real multiples of $u$ onto itself and the map $r_uf$ is the identity on the subspace $Ru$.'
Can anyone explain in detail to me why the map $r_uf$ is the identity on the subspace $Ru$?
Why do we have $$r_uf(u)=u$$ Can anyone explain why the last equality holds?