In the Jacobi algorithm, we start with a matrix $\Sigma \in \mathbf{S}_n$ and construct a sequence of symmetric matrices $\Sigma^{(m)}$, whose coefficients are denoted $\sigma_{ij}^{(m)}$, as follows:
\begin{itemize}
  \item We set $\Sigma^{(0)} = \Sigma$.
  \item When $\Sigma^{(m)}$ is known, we choose a pair $(p_m, q_m)$ with $p_m < q_m$.
  \item We then apply the calculations from Part 2 to the matrix $S = \Sigma^{(m)}$ and the pair $(p,q) = (p_m, q_m)$: we form the matrix $S'$ studied in this part, and call it $\Sigma^{(m+1)}$.
\end{itemize}

We define
$$R_m = \sum_{i,j=1}^{n} \left|\sigma_{jj}^{(m)} - \sigma_{ii}^{(m)}\right|, \quad \varepsilon_m = \sum_{i=1}^{n} \left|\sigma_{ii}^{(m+1)} - \sigma_{ii}^{(m)}\right|$$

Verify that $R_{m+1} - R_m \geqslant 2\varepsilon_m$. Deduce that the series $\sum_{m=1}^{\infty} \varepsilon_m$ is convergent.