I want to call a matrix a Jacobi matrix (cause there may be different notions of Jacobi matrices) if it is a tridiagonal matrix with positive off-diagonal entries. Now, I read that the spectrum of such matrices is simple and interlacing. Although, I find quite many proofs of the fact that the spectrum is interlacing, I could not see that it is simple. Just in one paper, it was said that this would be an immediate consequence of the tridiagonal form of the linear system $$(A - \lambda I)v=0.$$

Thus, now it should be somehow possible to conclude from this that for such Jacobi matrices the nullspace is one-dimensional, but I don't see how.