# Show that A** subset of A*, where A is a regular language

I do not understand the proof for this. I know that every word in $(A^*)^*$ is made up of words from $A^*$, and that this is made up from words in $A$. But how does this help with showing that $(A^*)^*$ is a subset of $A^*$.

• Have you tried applying the definition of the Kleene star on both sides? – Raphael Aug 26 '13 at 8:28

Knowing that $A$ is regular is not relevant.

Hint. Just replace "made up of words of $A^*$" by "is a concatenation of words of $A^*$" in your sentence and you will have the answer.

• Sorry but this still just doesn't make sense to me. How does knowing that A** is a concatenation of words of A* tell me that A** is a subset of A*? – user678392 Aug 25 '13 at 13:41

A concatenation of words of $A^*$ gives a word which is in $A^*$, because of the very definition of $A^*$. This gives the result.

Moreover, one can prove that $^*$ is a closure operator, which includes this property.