Prove $⌊2x⌋+⌊2y⌋≥⌊x⌋+⌊y⌋+⌊x+y⌋$ Prove the following for all real $x$
i. $⌊2x⌋+⌊2y⌋≥⌊x⌋+⌊y⌋+⌊x+y⌋$
ii. $⌊x⌋-2⌊x/2⌋$ is equal to either $0$ or $1$
For ($ii$.) I attempted to split it into cases of whether the fraction part {$x$} is $≥.5$ or $<5$ but that ended up being too tedious and I know there must be a more elegant, simpler method.
For ($i$) I tried cases like in part ($ii$) but since there are 2 variables that would lead to 4 cases.
An elegant and easy solution will be much appreciated

Part ii:
In general: $$\lfloor z\rfloor \leq z< \lfloor z\rfloor+1.$$ 
For ii, set $z=x/2$ then we get:
$$\left\lfloor\frac{x}{2}\right\rfloor\leq \frac{x}{2}<\left\lfloor\frac{x}{2}\right\rfloor +1$$
Double and you get:
$$2\left\lfloor\frac{x}{2}\right\rfloor\leq x<2\left\lfloor\frac{x}{2}\right\rfloor +2$$
Taking the floor gives:
$$2\left\lfloor\frac{x}{2}\right\rfloor\leq\lfloor x\rfloor<2\left\lfloor\frac{x}{2}\right\rfloor +2$$
Subtracting gets:
$$0\leq \left\lfloor x\right\rfloor-2\left\lfloor\frac{x}{2}\right\rfloor<2.$$
Which is the result you want.
Part i
You can actually prove the same way  that $\lfloor x+y\rfloor-\lfloor x\rfloor -\lfloor y\rfloor$ is always either $0$ or $1.$ More specifically, you can show that $\lfloor x+y\rfloor-\lfloor x\rfloor -\lfloor y\rfloor=1$ if and only if $\{x\}+\{y\}\geq 1.$ 
In particular, when $y=x,$ $\lfloor 2x\rfloor -2\lfloor x\rfloor=1$ if and only if $\{x\}\geq \frac{1}{2}.$
Now, if $\{x\}+\{y\}\geq 1$ then one or both of $\{x\}$ and $\{y\}$ are $\geq \frac{1}{2}$, so one or both of $\lfloor 2x\rfloor -2\lfloor x\rfloor$ and $\lfloor 2y\rfloor-2\lfloor y\rfloor$ are one. 
So that means, doe any $x,y:$
$$\lfloor x+y\rfloor -\lfloor x\rfloor -\lfloor y\rfloor\leq \left(\lfloor 2x\rfloor -2\lfloor x\rfloor\right)+\left(\lfloor 2y\rfloor -2\lfloor y\rfloor\right)$$
This is because when the left side is $0,$ we know the right side is at least $0,$ and when the left side is $1$, then the right side is either $1$ or $2.$
Adding $2\lfloor x\rfloor + 2\lfloor y\rfloor$ to both sides gives you:
$$\lfloor x+y\rfloor +\lfloor x\rfloor +\lfloor y\rfloor\leq \lfloor 2x\rfloor +\lfloor 2y\rfloor$$
