Prove that f (n) = n

Question

An induction problem

Given $f:\Bbb{N} \rightarrow \Bbb{N}\,$ That satisfies all of the stated conditions.

1. $ f(m*n)=f(m)*f(n)$
2. $f(2) = 2$
3. $m>n \rightarrow f(m) > f(n)$

I need to prove that $\forall\,n\in\,\Bbb{N}\quad\,f(n) = n$

Answer 1

$$f(1 * 1)=f(1) * f(1) \implies f(1)=1,$$ next $$f(2 * 2)=f(2) * f(2) \implies f(4)=4,$$ next $$f(4)> f(3) > f(2) \implies f(3)=3.$$ Can you proceed? Hint: Use strong induction!

Answer 2

In case you or someone else gets stuck at some step.

$4\geq f(4)>f(3)>f(2)=2$. Therefore, $f(3)=3$.

Assume that $p>2$ is prime and for all primes $q<p$ we have $f(q)=q$. In particular $f(n)=n$ for all $n$ divisible only by primes $<p$.

Let $q$ be the next prime after $p$. Then $q+1=2\frac{q+1}{2}$ and $\frac{q+1}{2}$ is $<q$ and therefore divisible only by primes $<p$. Therefore, $f(p+1)=f(2\frac{q+1}{2})=f(2)f(\frac{q+1}{2})=2\frac{q+1}{2}=q+1$.

It follows that $q+1=f(q+1)>f(q)>f(q-1)=q-1$, where the last equality is due to $q-1$ being divisible only by primes $<p$. Therefore $f(q+1)=q+1$.

By induction on the primes $f(q)=q$ for all primes. Since the function is completely multiplicative, then $f(n)=n$ for all $n\in\mathbb{N}$.