Consider the set $$\Bbb R^n :=\{x=(x_1,...,x_n):x_1,...,x_n \in \Bbb R \}.$$
For $x,y\in \Bbb R^n$, we define $<$ as below: $$ x<y \iff \exists j \in \{1,..,n \} \left( x_j<y_j \wedge \forall i \in \Bbb N (i<j \to x_i=y_i)\right).$$
The question is:
if $j=n$,
follows $(x_1=y_1),...,(x_{n-1}=y_{n-1})$,
or do they differ?
The ordering you describe is the lexicographic ordering, the one that allows you to say that "car" will appear before "cat" in a dictionary. In this example, $j=3$ (i.e. the words differ only at their last letter, which is the third), and you are right, this means that the $n-1$ previous letters are the same.