Proof:

From corollary (36.4) the case $n = 1$ follows (exercise 1). The general case is done by induction. Let us consider the covering $\{U, V\}$ where $U = S^n - \{{\bf e}_{n+1}\}$, and $V = S^n - \{-{\bf e}_{n+1}\}$. The map $R_n$ fixes $U$ and $V$ but when restricted to the equator $S^{n-1}$ gives $R_{n-1}$. The naturality of the Mayer Vietoris sequence gives us the commutative diagram

$\begin{CD} H_n(U\cup V) @> {\delta_n} >> H_{n-1}(U\cap V) @> {H_{n-1}(r)}>> H... ...> \delta_n >> H_{n-1}(U\cap V) @> {H_{n-1}(r)}>> H_{n-1}(S^{n-1}). \\ \end{CD}$

The map $H_{n-1}(r)$ is isomorphism induced by the retraction of $U\cap V$ onto the equator $S^{n-1}$. The connecting homomorphisms $\delta_n$ are isomorphisms as we have seen in the last lecture. Since the map $H_{n-1}(R_{n-1}):\mathbb{Z}\longrightarrow \mathbb{Z}$ on the extreme right is given by multiplication by $-1$, the same is the case with the map $H_n(R_n)$ on the extreme left whereby we conclude that $R_n$ has degree $-1$.