  The first step when solving an infinite-dimensional eigenvalue problem is
often to discretize it. We show that one must be extremely careful when
discretizing nonlinear eigenvalue problems. Using examples, we show that
discretization can: (1) introduce spurious eigenvalues, (2) entirely miss
spectra, and (3) bring in severe ill-conditioning. While there are many
eigensolvers for solving matrix nonlinear eigenvalue problems, we propose a
solver for general holomorphic infinite-dimensional nonlinear eigenvalue
problems that avoids discretization issues, which we prove is stable and
converges. Moreover, we provide an algorithm that computes the problem's
pseudospectra with explicit error control, allowing verification of computed
spectra. The algorithm and numerical examples are publicly available in
$\texttt{infNEP}$, which is a software package written in MATLAB.
