Let $D = \operatorname{Diag}(\alpha_{1}, \ldots, \alpha_{d})$ be a diagonal matrix with positive coefficients and let $R \in \mathrm{O}_{d}(\mathbb{R})$.
\begin{itemize}
\item[(a)] Show that for all $1 \leqslant i \leqslant d$, we have $|R_{ii}| \leqslant 1$ where $R_{ii}$ is the $i$-th diagonal coefficient of $R$.
\item[(b)] Deduce that $\langle D, R \rangle \leqslant \operatorname{tr}(D)$.
\end{itemize}