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 $\sigma_{ij}^{(m)}$. We decompose $\Sigma^{(m)}$ in the form $D^{(m)} + E^{(m)}$ where $D^{(m)}$ is diagonal and $E^{(m)}$ has zero diagonal.
Show that the sequence $\left(D^{(m)}\right)_{m \in \mathbb{N}}$ is convergent. We denote its limit by $D$.