I am wondering why is $\{\frac{y}{n} + k\} = \{\frac{y}{n}\}$ where $k$ is an integer. $\{x\}$ is a fractional part function where $\{x\} = x - \lfloor x\rfloor$. I know it makes sense logically, but when I try to prove it I can not seem to get rid of $k$.
$\{\frac{y}{n} + k\} = \frac{y}{n} + k - \lfloor\frac{y}{n} + k\rfloor$. How does $k$ disappear from this equation?
First of all, in general, $\{\frac yn+k\}\neq \{\frac yn\}$. For example, $y=0,n=1,k=\frac12$ are a counterexample. But yes, the equality holds if $k$ is an integer. Then, it can even be written more simply as $\{x+k\}=\{x\}$, where $x$ can be any real number.
To actually prove that equality, you can use the following hint:
The function $\lfloor x\rfloor$ has the property that $\lfloor x+n\rfloor = \lfloor x \rfloor + n$ for all $n\in\mathbb N$.