I was recently asked to help someone with the following question on their first year analysis course.
Recall that $\mathbb{N}$ is the set of all positive integers. Use the principle of induction to show that $n \ge 1$ for all $ n \in \mathbb{N}.$ [ Hint: Let $ S = \lbrace n \in \mathbb{N} | n \ge 1 \rbrace.$ ]
I am concerned that the first sentence of the question assumes that we already know exactly what the positive integers are, which renders the rest of the question redundant.
I would prefer to see the question written:
Recall that $\mathbb{N}$ is the set of $ x \in \mathbb{R}$ such that $x$ is a member of every inductive set. [ $P$ is an inductive set if (a) $ 1 \in P $ and (b) $ x \in P \Rightarrow x+1 \in P.$ ] Then we can answer as follows:
With $S$ as in the hint, by definition $ S \subseteq \mathbb{N}.$
$1 \in S$ since $1 \in \mathbb{N},$ as it is an inductive set. For $ n \in S, $ $n+1 > n \ge 1,$ therefore $n+1 \in S.$ Hence $S$ is an inductive set, and so $ \mathbb{N} \subseteq S.$ Therefore $ S=\mathbb{N}.$ Therefore $1$ is the least element of $\mathbb{N}.$
Am I being way too picky, or is the question flawed as it stands?