If $f: \mathbb{N} \to \mathbb{Z}$ defined as $f(n)=n^{2007}-n!$
Then Prove that it is an Injective function
My try:
According to the definition of Injective function:
If $p,q \in \mathbb{N}$ and if $$f(p)=f(q)$$ Then we need to Prove that $p=q$
We have:
$$p^{2007}-p!=q^{2007}-q!$$
$$p^{2007}-q^{2007}=p!-q!$$
Without loss of generality let $p \gt q$ and $p=q+m$
We have:
$$(q+m)^{2007}-q^{2007}=q!\left((q+m)(q+m-1)(q+m-2)\cdots (q+1)-1\right)$$
By Binomial Theorem we have
$$\binom{2007}{1}q^{2006}m+\binom{2007}{2}q^{2005}m^2+\cdots+m^{2007}=q!\left((q+m)(q+m-1)\cdots (q+1)-1\right)$$
Any way to proceed here?