  We show existence of a unique solution and a comparison theorem for a
one-dimensional backward stochastic differential equation with jumps that
emerge from a L\'evy process. The considered generators obey a time-dependent
extended monotonicity condition in the y-variable and have linear
time-dependent growth. Within this setting, the results generalize those of
Royer (2006), Yin and Mao (2008) and, in the $L^2$-case with linear growth,
those of Kruse and Popier (2016). Moreover, we introduce an approximation
technique: Given a BSDE driven by Brownian motion and Poisson random measure,
we consider BSDEs where the Poisson random measure admits only jumps of size
larger than $1/n$. We show convergence of their solutions to those of the
original BSDE, as $n \to \infty.$ The proofs only rely on It\^o's formula and
the Bihari-LaSalle inequality and do not use Girsanov transforms.
