I have started studying logic from this source. I was trying the problems and got stuck on the following one:
Suppose that $n \in \mathbb{W}$ and that $n\gt 1$. Prove that $\exists$ infinitely many positive whole numbers that are not whole number multiples of $n$.
I tried the following:
Let $a$ be non-whole number multiple of $n$. By Euclid's Lemma we have $a=nq+r$ with $r$ satisfying the stronger inequality $0\lt r \lt n$. Clearly every number of the form $n(q+k)+r$ for $k\in \mathbb{Z}^+$ can be rewritten as $nq'+r$. Since the cardinality of $\mathbb{Z}^+$ is $\infty$ therefore there must be infinitely many $a$ satisfying the condition. Hence, proved.
Is this correct? Is there any discrepancy in the proof? I would also like to get an explanation from the perspective of formal logic. Thanks!
If $\ n>1\ $ is an integer , the (infinite many) positive integers of the form $$kn+1$$ where $\ k\ $ is a positive integer , are obviously not a multiple of $\ n\ $, otherwise we would have $\ n\mid 1\ $ and therefore $\ n=1\ $.