  This paper addresses two different but related questions regarding an
unbounded symmetric tridiagonal operator: its self-adjointness and the
approximation of its spectrum by the eigenvalues of its finite truncations. The
sufficient conditions given in both cases improve and generalize previously
known results. It turns out that, not only self-adjointness helps to study
limit points of eigenvalues of truncated operators, but the analysis of such
limit points is a key help to prove self-adjointness. Several examples show the
advantages of these new results compared with previous ones. Besides, an
application to the theory of continued fractions is pointed out.
