How can I prove that Hamming distance is upper bound for Levenshtein distance?

Question

We have a spellchecker software. And one of it crucial parts is hypothesis generator which use Levenshtein distance as a measure of distance between words. The problem with Levenshtein distance is that it's not so easy to calculate (from application performance point of view), so we are came up with an idea to use Hamming distance as a fast reject algorithm for candidates.

So I interested in formal proof of the following statement: suppose you have two strings of equal length. Is it right that Levenshtein distance between them is no more that Hamming distance?

Answer 1

Homework, I bet.

Suppose $a$ and $b$ are strings of equal length with Hamming distance $k$. This means the strings differ in $k$ places, say $a_{i_1} \neq b_{i_1}$, ..., $a_{i_k} \neq b_{i_k}$. It takes $k$ edits to change $a$ into $b$, namely replacement of the characters $a_{i_1}, \ldots, a_{i_k}$ by the corresponding $b_{i_1}, \ldots, b_{i_k}$. Therefore the Levenshtein distance between $a$ and $b$ is no more than $k$. But it could be smaller than $k$, of course.