Don’t understanding argument on words of a certain form
I think the book is right.  In your example, the word $u=abcd$ satisfies the conditions: it is not a prefix of a word in $a^* b^* c^*$.

In general, I think that proof is valid.  I'll try to explain why the minimality of $n$ proves the existence of such a word $u \in L$. If there did not exist such a word $u \in L$, then that would mean that every word $u \in L$ is a prefix of $l_1^* \cdots l_{n-1}^*$, which in turn would mean that $n$ wasn't the minimum possible (we could have used $n-1$), which contradicts how $n$ was chosen in the second sentence.  That's a contradiction.  If you start from an assumption and derive a contradiction, then you can conclude that the assumption must have been wrong.  In particular, the only alternative that remains possible is that there does exist such a word $u \in L$.

Put another way, there are only two possibilities: either there does exist such a word $u \in L$, or there doesn't.  I've shown why the second choice is always impossible.  That means the first choice must actually always be true.