By assumption our points $u^i(z_i)$ don't run away along the semi-infinite ends attached to $\Sigma'$. The *monotonicity lemma* should now prevent the sequence of renormalized holomorphic discs from "thinning" out and having noncompact image. That is, our maps must use up at least a certain amount of energy
for every ball whose center it passes through.

Restating, per the comments: The boundedness assumption means precisely that the points don't run away from $\Sigma'$, i.e. they are stuck in $\Sigma'$ union a collar $\partial\Sigma'\times[0,k]$ and cannot escape along $\partial\Sigma'\times[0,\infty)$, otherwise you're in Case#1 (in the paper) which has the points running away along the neck. As for the rest of the images of the maps, given a ball centered around any point that's in your map's image, monotonicity prevents this map from having arbitrarily small area in the ball.