Can someone please explain the following proof:
that if $u$, $v$ are sequences of length $n$ over any alphabet such that $d(u, v) = a + b$, then there always exists a sequence $z$ such that $d(u, z) = a$ and $d(z, v) = b$.
So I understand that idea of how to find the distance between $u$ and $v$ and understand how it can be $a + b$, however I cant understand how a new random sequence $z$ exists to satisfy the other two conditions in relation. How can adding a new sequence '$z$' give the result of '$a$' and '$b$' separately?
Thanks in advance to anyone who helps explain this proof to me.
Distance is defined as "differs in so-and-so many positions". Pick $a$ of the positions where $u$ and $v$ differ and adjust them.