This is a review for a midterm, so I have the solution but I do not understand some key components. Any insight would be greatly appreciated.
$f(m) = (\lfloor p^{(\frac{1}{m})}\rfloor-1)\cdot p^{(\frac{1}{m})}$
where $p$ is a fixed prime number, $m\in\mathbb{N}$.
Construct a set $ S = \{m_{1}\cdot f(1) + m_{2}\cdot f(2) + m_{3}\cdot f(3) + \ldots \;|\; m_{i}\in\mathbb{Z}\} $
Is $S$ a countable set?
Proof: Notice that as $m$ approaches infinity $p^{(\frac{1}{m})}$ approaches $1$. Then there must exist $M\in \mathbb{N}$ such that $1 < p^{(\frac{1}{m})} < \frac{3}{2}$ for all $m \geq M$.
Why is the upper-bound $\frac{3}{2}$?
However $f(m) = 0$ for all $m \geq M$. Hence $S$ is an image of $\mathbb{Z}^{M}$.
What does that mean, $S$ is an image of $\mathbb{Z}^{M}$? Why is that significant?
Mapping: $ (m_{1},\ldots,m_{M}) \mapsto (m_{1}\cdot f(1),\ldots,m_{M}\cdot f(M)) $
Where does this mapping come from?
Thus $S$ is a countable set.
What is the most common way to prove that something is a countable set?