By assumption our points $u^i(z_i)$ don't run away along the semi-infinite ends attached to $\Sigma'$. The *monotonicity lemma* now prevents 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: The boundedness assumption means precisely that our bubble points 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 of $\Sigma^{R_i}$. As for the rest of the images of the maps, given balls centered around various points in your maps' image, monotonicity prevents the maps from having arbitrarily small area in any of these balls and hence the images of the maps are contained in some overall compact region.