I can't seem to prove propositions involving floor/ceiling function and the like.. What I know is: The floor function of $x$, denoted $[x]$, is the greatest integer smaller than $x$, ceiling of $x$ is the smallest integer greater than $x$.
We define $\{x\} = x- [x]$, the factional part of x.
Till now all is okay and fine, but whenever I come across a proving question involving these type of functions, I can't even seem to get started.
I try to use extreme cases to disprove the statement if possible, but when it comes to proving, I'm just....lost.
Like where do I even begin??
An example would be:
Proposition: For any real numbers $x, y,$ and $z$, prove that
$[x + y + z] = [x + y] + [z + \{x + y\}]$.
I don't even know what to do with this except try and check it for some cases, which I already did.

Note that $$\tag1a=\lfloor a\rfloor+\{a\}$$ by definition. Also, if $n$ is an integer, then
$$ \tag2\lfloor n+a\rfloor = n+\lfloor a\rfloor.$$
In order to show
$$ \lfloor x+y+z\rfloor = \lfloor x+y\rfloor +\lfloor z+\{x+y\}\rfloor,$$
use $(2)$ with substitution $n\leftarrow\lfloor x+y\rfloor$, $a\leftarrow z+\{x+y\}$ and use $(1)$ with substitution $a\leftarrow x+y$.
