Let c:$\mathbb{N}\rightarrow\mathbb{N}$ be a strictly monotonic inreasing function, proof by induction that $c \geq id_{\mathbb{N}}$ where id is the identity $\mathbb{N} \rightarrow \mathbb{N}:n \rightarrow n$. I have to do it with induction, but c could be any strictly monotonic increasing function, so how do I do that?
Induction proof: monotonic inreasing
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monotone-functions
1 Answers
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We have that $c(1)$ is a natural number and thus $c(1)\geq 1$. So we have $c(n)\geq n$ for $n=1$.
Suppose now that for some natural number $k$ it is $c(k)\geq k$. Based on this assumption we will prove that $c(k+1)\geq k+1$. Since $c$ is a strictly increasing function with integer values, we have:
$c(k+1)>c(k)\Rightarrow c(k+1)\geq c(k)+1\Rightarrow c(k+1)\geq k+1$
Therefore the inductive proof is over. Hence $c(n)\geq n\ \forall n.$
Note: If you include $0$ in the set of natural numbers, the proof is similar.