Compression

For any given finite string, there is a compression method (an invertible (one-to-one) function) for which the finite string is compressed to the empty string.

**Proof:**

Given a finite string i, we can construct the one-to-one function \( C_i \): $$ C_i(x) = \begin{cases} \epsilon &\mbox{if } x = i \\ i &\mbox{if } x = \epsilon \\ x &\mbox{otherwise} \end{cases} $$ which means that $$C_i(i) = \epsilon$$ i.e. the input string \( i \) is mapped to the empty string \( \epsilon \), and $$C_i(\epsilon) = i$$ i.e. the empty string \( \epsilon \) is mapped to \( i \), and for all other strings $$C_i(x) = x$$ i.e. we have the identity mapping.