Find the value of $\lfloor x+y \rfloor$ where $x \in \mathbb{R}$, $y \in \mathbb{Z}$

I'm trying to understand properties of the greatest integer function and I am struggling to find the value of $\lfloor x+y \rfloor$ where $x \in \mathbb{R}$, $y \in \mathbb{Z}$, and prove that it is correct value.

I don't really know how to prove this, but I have been dividing it into different cases. I think that it equals $\lfloor x \rfloor + y$ when $x,y$ are both positive but not sure how to prove it. Depending on if one or both $x$ and $y$ are negative, and their ultimate sum, I get different values. I am having trouble determining when exactly this happens though and then proving the results. Any help would be great, thanks!

It is always $\lfloor x \rfloor + y$. Write $x = \lfloor x \rfloor + \{ x \}$ where $0 \le \{ x \} \lt 1$. Then you just throw away the $\{ x \}$ because the rest is the integer. Remember that $\lfloor -2.3 \rfloor = -3$, not $-2$

Note, lets use $x$ as the real and $n$ as the integer.

Since $x - 1 \le \lfloor x \rfloor \le x$, it follows that $-x \le - \lfloor x \rfloor \lt -x+1.$

Combining this inequality with $x+n - 1 \lt \lfloor x+n \rfloor \le x + n$, we obtain $n-1 \lt \lfloor x+n \rfloor - \lfloor x \rfloor \lt n+1.$

Hence $\lfloor x+n \rfloor - \lfloor x \rfloor = n.$