Strictly decreasing function $f(n)>f(n+1)$ is not definable in $\mathbb{N}$ (Set Theoretic (Neumann's) construction of $\mathbb{N}$). I'm studying some Set Theory now and I oppose to a problem which I guess it is related to well ordering of Natural numbers.
The problem is:
Prove that there is no function $f:\mathbb{N}\longrightarrow\mathbb{N}$ which for all $n\in\mathbb{N}$ we have $f(n+1) < f(n)$.
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Maybe we answer it as below:
If there is any, we should have $f(0)>f(1)>f(2)>f(3)>\dots$ and it's not the case because the set of Natural numbers is not "left-unbounded".
but the MAIN problem is I must use the set theoretic construction of Natural numbers (Neumann's construction which is based on $\varnothing$ set. ($0=\varnothing, 1=\{\varnothing\}, 2=\{\varnothing,\{\varnothing\}\},\dots$)).
How can I describe well-ordering in $\mathbb{N}$ or what do I should to do?
Thank you in advance.
 A: Suppose you know that $\mathbb{N}$ is well-ordered by $<$. This means that every nonempty subset of $\mathbb{N}$ has a least element. Now, $\{ f(n)\, :\, n \in \mathbb{N} \}$ is certainly a non-empty subset of $\mathbb{N}$, and so it has a least element. Is this possible if $f$ is strictly decreasing?
If you need to show that $\mathbb{N}$ is well-ordered, you'll first need to tell me what set-theoretic construction you're using! (I've removed the answer I previously left about von Neumann's construction etc.)

Added: You can prove by induction that $\mathbb{N}$, as constructed in your post, has no infinite descending $\in$-chains. First note that if $n \in \mathbb{N}$ then $n$ is a natural number, so it suffices to show that no natural number $n$ has an infinite descending $\in$-chain.
The base case is just $\varnothing$: the length of the longest $\in$-chain is $0$ since it has no elements at all!
Now suppose that all $\in$-chains in the set $n$ have length $\le n$. (That is, if $a_1, a_2, \dots, a_k \in n$ and $a_1 \in a_2 \in \cdots \in a_k$ then $k \le n$.) Prove from this that all $\in$-chains in the set $n+1 = n \cup \{ n \}$ have length $\le n+1$.
