  A Bernoulli Gibbsian line ensemble $\mathfrak{L} = (L_1, \dots, L_N)$ is the
law of the trajectories of $N-1$ independent Bernoulli random walkers $L_1,
\dots, L_{N-1}$ with possibly random initial and terminal locations that are
conditioned to never cross each other or a given random up-right path $L_N$
(i.e. $L_1 \geq \cdots \geq L_N$). In this paper we investigate the asymptotic
behavior of sequences of Bernoulli Gibbsian line ensembles $\mathfrak{L}^N =
(L^N_1, \dots, L^N_N)$ when the number of walkers $N$ tends to infinity. We
prove that if one has mild but uniform control of the one-point marginals of
the lowest-indexed (or top) curves $L_1^N$ then the sequence $\mathfrak{L}^N$
is tight in the space of line ensembles. Furthermore, we show that if the top
curves $L_1^N$ converge in the finite dimensional sense to the parabolic
Airy$_2$ process then $\mathfrak{L}^N$ converge to the parabolically shifted
Airy line ensemble.
