Apostol's Calulus: Prove that $[x+y] = [x]+[y]$ or $[x]+[y]+1$, where $[·]$ is the floor function. 
Prove that $[x+y] = [x]+[y]$ or $[x]+[y]+1$, where $[·]$ is the floor
  function

I'm Having a little bit of trouble with the last part of this proof.
First, I will use the definition of floor function:
$[x] = m ≡ m ≤ x < m+1$ 
and 
$[y] = n ≡ n ≤ y < n+1$
so, $[x]+[y] = m+n ≡ m+n ≤ x+y < m+n+2$
This is where I get stuck; I have that $[x+y] = t ≡ t ≤ x < t+1 $, so putting 
$m+n ≤ x+y < m+n+2$ in that form, seems impossible, let alone the one that corresponds to $[x]+[y]+1$.
Could you help me with this last part? 
Thanks in advance.

Alternatively.  By definition $[x+y]$ is the largest possible integer that this less than or equal to $x+y$.  But $[x] \le x$ and $[y] \le y$ so $[x] + [y] \le x+y$.  So $[x]+[y] \le [x+y]$.  
Likewise $[x+y] + 1$ by definition is the smallest possible integer that is larger $x + y$.  But $[x]+ 1 > x$ and $[y] + 1 > y$ so $[x]+[y] + 2 > x+y$.  So $[x] + [y] + 2\ge [x+y]+1$.
So $[x]+[y] + 1 \ge [x+y]$.
So $[x]+[y] \le [x+y] \le [x] + [y] + 1$
As $[x]+[y]$ and $[x+y]$ and $[x] + [y]+1$ are all integers. And there is NO integer between $[x] +[y]$ and $[x]+[y]+1$ there are only two options $[x] + [y]= [x+y]$ or $[x+y] = [x] + [y]+1$.
.......
And a third way.
$[x] \le x < [x]+1$ means $0 \le x - [x] < 1$.
A) $0 \le x-[x] < 1$ and $0 \le y -[y] < 1$ so $0 \le x+y -[x]-[y] < 2$.
B) $0 \le x+y - [x+y] < 1$.
Reverse B) to get
B') $-1 < [x+y] - x - y \le 0$.
Add B' and A to get:
$-1 < ([x+y] - x - y) + (x + y - [x]-[y]) < 2$ so
$-1 < [x+y] - [x] -[y] < 2$ or 
$0 \le [x+y] -[x] -[y] \le 1$ or 
$[x]+[y] \le [x+y] \le [x] + [y] + 1$.
.....
Basically:  $x+y$ is between $[x+y]$ and $[x+y] + 1$; two integers that are only $1$ apart.
But $x + y$ is also between $[x] + [y]$ and $[x] + [y] + 2$; two integers that are only $2$ apart. 
There are only so many choicese to find these integers $[x+y], [x]+[y], [x+y] + 1$ and $[x]+[y] + 2$ so that they all fit in such a tight range.  
It doesn't matter how you prove it but you must have $[x+y]$ and $[x]+ [y]$ within one of each other and you must have $[x]+[y]\le [x+y]$.  There are only two ways that can happen.
