So here's the question:
Prove or disprove: For every $x \in \mathbb{Q}$, there is a unique $n \in \mathbb{N}$ which is the closest natural number to $x$.
I know we can define a rational number as $\frac{a}{b}$, with $a,b$ as natural numbers, but I'm not sure where to go from there. Any hints/help would be greatly appreciated!
No, there is not always a unique closest natural number, as mentioned in comments. In more detail:
If $q$ is a rational number of the form $q= a/2$ with $a$ an odd positive integer, then the two natural number $(a-1)/2$ and $(a+1)/2$ are both at distance $1/2$ from $q$. Every natural number other than these two is either less than $(a-1)/2$ or greater than $(a+1)/2$. Thus its distance to $q$ is strictly greater than $1/2$.
Therefore, the smallest distance from a natural number to such a $q$ is $1/2$. This distance is however attained by two natural numbers, $(a-1)/2$ and $(a+1)/2$, and therefore there is not always a unique closest natural number.