I'm looking for some insight on how to formalize my ideas for this proof.
So because the sequence is null, we know that the sequence taper off to $0$ for some $n$ values beyond $N$. We can therefore use $N$ to divide the sequence into a finite, non-empty part (the part before $N$) and an infinite, bounded part (the part after $N$).
There is a theorem stating that every finite, non-empty set has a maximum, so we know that the part of the sequence before $N$ has a maximum. Since we know that the part after $N$ tapers off to $0$, we know that this maximum value applies to this part of the sequence too. Thus, every positive, null sequence has a maximum.