Well-ordering the set of all finite sequences. Let $(A,<)$ be well-ordered set, using <, how can one define well-order on set of finite sequences? 
(I thought using lexicographic order) 
Thank you! 

The lexicographic order is fine, but you need to make a point where one sequence extends another -- there the definition of the lexicographic order may break down. In this case you may want to require that the shorter sequence comes before the longer sequence.
Generally speaking, if $\alpha$ and $\beta$ are two well-ordered sets, then we know how to well-order $\alpha\times\beta$ (lexicographically, or otherwise). We can define by recursion a well-order of $\alpha^n$ for all $n$, and then define the well-order on $\alpha^{<\omega}$ as comparison of length, and if the two sequences have the same length, $n$, then we can compare them by the well-order of $\alpha^n$.
