I'm having trouble with the following exercise:
Let $\Sigma = \{a,b,c\}$ and $L$ be a formal language, that consists of all words which contain all three letters at least once. Show that $L$ is infinite.
I figured out that I could define a word
$w_0 = abc, w_0\in L$
so the condition is met, and then say
$w_i = w_0\cup_{i = 1}^{\infty}a, w_i \in L$
..which shows that you could add infinitely many letters(a's in this case) to it and therefore you can get infinitely many words.
Now I don't know how to show that the rule above does indeed produce infinitely many words, isn't it just obivous? I'm also not sure about the notation - should I just add "This shows that L is infinite" ? And finally I don't know if I'm allowed to use a specific pre-condition like "abc"