I am looking for a proof verification for the question in the title: Probability that a randomly selected integer $n\ge0$ is divisible by $p>0$ tends to $1/p$ for large $n$. I want to get the wording and notation accurate and precise as possible so any comments or suggestions are appreciated. (I believe we assume $p\in\mathbb{N}$). My attempt is below.
For $n\in\mathbb{N_0}$ we can write $n=p\cdot m+r$ for $p\in\mathbb{N}$ and $m,r\in\mathbb{N_0}$. Here $m$ is the number of integers less than $n$, which is also divisible by $n$, and $r$ is the remainder $0\le r\le p-1$. Then the probability is given by $$P(n\text{ is divisible by }p)=\frac{m}{n}=\frac{1}{n}\cdot\frac{n-r}{p}=\frac{1}{p}\left(1-\frac{r}{n}\right)$$ which tends to $1/p$ for fixed $r$ as $n\to\infty$.