Show that the following construction is not a correct proof for Context Free Grammars
$ S\to aSb \mid \varepsilon $.

Hint (due to Klaus Draeger): Consider what happens when you apply transformation (ii) to this grammar. Does the resulting grammar $G'$ satisfy $L(G')=L(G)^*$?

Explanation: The point is that the $S\to SS$ production can be used "inside" the grammar, not just on the "top-level" where it is intended. We can derive $S\to^* aSSb \to^* aababb$ which is of the wrong form. (And the grammar has $S\to\varepsilon$ anyhow.)