If A and B are finite sequences and A , B = B , A...

Step 1
If A,B are strings of length $|A|,|B|\le 1$ and $AB=BA$ then there is some string D such that $A=D^k,B=D^l$ with $k,l\in \{0,1,..\}$
Suppose that when A,B are strings of length $|A|,|B|\le n$ and $AB=BA$ then there is some string D such that $A=D^k,B=D^l$ with $k,l\in \{0,1,..\}$.
Step 2
Now suppose $|A|\le n+1,|B|=n+1$ and $AB=BA$. If $|A|=|B|$ then $A=B$ and we can take $D=A$, so suppose $|A|\le n$. Then we must have $B=AC$ for some $|C|\ge 1$. Then $AAC=ACA$ and so $AC=CA$ and hence there is some D such that $A=D^k,C=D^l$. Then $A=D^k,B=D^{k+l}$.