One of my tasks is to proof that something is correct or incorrect using induction. Since I am from Germany and don't know the right word in English I do my best to give all necessary info. We are talking about "words" and amounts of words which are in so called alphabets.
Let $\Sigma \overset{\Delta} = \{a, b\}$ and $S_1 \overset{\Delta} = \{a^n \mid n \in \Bbb N\}$. Prove using induction that $$\forall w \in \Sigma^* . w \in S_1 \to \big( w = \lambda \lor (\forall \ 1 \le i \le |w| . (w)_i = a) \big) .$$
How to prove that? Any help is upvoted, thank you.
This is obvious, there is nothing to prove here: if a word belongs to $S_1$, then it is either the empty word, or it is a concatenation of "$a$"s, by the mere definition of $S_1$.