  The (abstract) Cuntz algebra is generated by non-unitary isometries and has
therefore no intrinsic finiteness properties. To approximate the elements of
the Cuntz algebra by finite-dimensional objects, we thus consider a spatial
discretization of this algebra by the finite sections method. For we represent
the Cuntz algebra as a (concrete) algebra of operators on a Hilbert space and
associate with each operator in this algebra the sequence of its finite
sections. The goal of this paper is to examine the structure of the
$C^*$-algebra which is generated by all sequences of this form. Our main
results are the fractality of a suitable restriction of this sequence algebra
and a necessary and sufficient criterion for the stability of sequences in the
restricted algebra. These results are employed to study spectral and
pseudospectral approximations of elements of the Cuntz algebra.
