# Question on a proof involving tightness and almost sure convergence of a sequence

I'm having a hard time understanding the proof of Lemma 17 in this article. Essentially, the assertion of the lemma boils down to replacing a constant in a sequence of random variables that satisfies a CLT (from the previous lemma) by a sequence that converges almost surely to said constant and obtaining that the CLT still holds (at least to my understanding).

However, I do not understand why the authors introduce the processes $A_{n,j}(v)$ in the proof in the first place and verify the tightness. The verification itself is clear, I just don't see why it is useful - it seems to be some kind of deeper (or maybe standard) result I am missing, since the authors seem to find it obvious that said tightness and almost sure convergence are sufficient for the statement of the lemma. Can someone point me in the right direction here and/or explain what I'm missing?