Suppose I have a question similar to:
Let $S$ be defined recursively by
(1) $5 ∈ S$ and
(2) if $s ∈ S$ and $t ∈ S$, then $st ∈ S$.
Let $A = \{5^i \mid i ∈ Z+\}$. Prove that $A ⊆ S$ by mathematical induction.
I know for mathematical induction you need a base case and then you want to assume the hypothesis and try to show that (n+1) would also be true. Maybe the recursive definitions are throwing me off, but I don't know where to begin.
The base case is $n = 1$. You need to prove that $5^1 \in S$.
The induction step: You need to prove that if $5^n \in S$ then $5^{n+1} \in S$.
If you do those, the it follows that all $5^n \in A$ are in S. And therefore $A \subset S$.
Can you do these?
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You have the rule that if $s, t \in S$ then $st \in S$. Do you have any values in $S$ that when multiplied together equal $5^{n+1}$?