Jacobi convergence for a symmetric positive-definite tridiagonal matrix

If $A=D+L+L^T$ is symmetric positive definite and tridiagonal, set $S=\mathrm{diag}(1,-1,1,-1,\dots )$. Thenis positive definite. Applying the Householder-John theorem to $M=D$ and $N=-(L+L^T)$ proves that the Jacobi iteration converges.