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Let $m, n \in \mathbb{N}$. If $n$ is divisible by $m$, then $m \le n$.

So far I have:

Let $m,n \in \mathbb{N}$ and assume that $n$ is divisible by $m$. Therefore, there exists $j \in \mathbb{Z}$ such that $n=jm$. By Proposition 2.11 (in the text our class uses), $j \in \mathbb{N}$. Therefore, by Proposition 2.21, $j \ge 1$.

Any help would be greatly appreciated, I don't even know if I'm going in the right direction here.

user3182252
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1 Answers1

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First, a Lemma: If $k $ is a positive integer and $m \in \mathbb{N}$, then $mk \geq m $. To show this, suppose by contradiction $mk < m $. Then $k < 1$ which is a contradiction. Hence $mk \geq m$.

Now, suppose $n$ is divisible by $m$, then can find $k$ positive such that $n = km $. BY the lemma, $km \geq m$. Hence, $n \geq m$ as required.

hope this helps