Let H be a separable Hilbert space with an orthonormal basis {en/n∈N}, T be a bounded tridiagonal operator on H and Tn be its truncation on span ({e1, e2, ... , en}). We study the operator equation Tx = y through its finite dimensional truncations T nxn = yn. It is shown that if {∥Tn-1en∥} and {∥Tn*-1en∥} are bounded, then T is invertible and the solution of Tx = y can be obtained as a limit in the norm topology of the solutions of its finite dimensional truncations. This leads to uniform boundedness of the sequence {Tn-1}. We also give sufficient conditions for the boundedness of {∥Tn-1en∥} and {∥Tn*-1en∥} in terms of the entries of the matrix of T. © 2005 Elsevier Inc. All rights reserved.