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**mathwizard** 1. Prove by induction:

(a) $\displaystyle 1 \le n\ \forall n \in N;$

(b) $\displaystyle \forall n \in N,$ either $\displaystyle n = 1 $ or $\displaystyle n-1 \in N$; and

(c) $\displaystyle \forall n \in N,$ there is no element $\displaystyle m$ of $\displaystyle N$ in the range $\displaystyle n < m < n+1.$

2. Prove by induction on $\displaystyle n$ that $\displaystyle \forall m, n \in N\ \exists q \in N$ such that $\displaystyle qm > n.$

Also prove that for every $\displaystyle n \in N$, there is an $\displaystyle m \in N $ such that $\displaystyle 2^m > n $.