Confusion in deduction of Hermite's Identity. I was following the proof of identity here. 
Before the proof, it wrote a relation $$\lfloor x\rfloor\le \left\lfloor x + \frac{1}{n}\right\rfloor \le \cdots \le \left\lfloor x + \frac{n-1}{n} \right\rfloor  
$$ & showed that each term can take two values $\lfloor x\rfloor$ & $\lfloor x\rfloor + 1$.
Now, can anyone explain me how the page wrote the above relation? And what does it mean that they can take only two values??

The number $\left\lfloor x + \frac 1 n \right\rfloor$ is necessarily at least as big as $\lfloor x\rfloor$ but may or may not be at least as big as $\lfloor x\rfloor + 1$, depending on what numbers $x$ and $n$ are.  Suppose, for example that $x=5.8$ and $n=12$.  Then
$$
\left\lfloor x + \frac 1 n \right\rfloor = \lfloor 5.88333\ldots\rfloor = 5 = \lfloor x\rfloor.
$$
However, suppose $x=5.8$ and $n=4$.  Then
$$
\left\lfloor x + \frac 1 n \right\rfloor = \lfloor 6.05\rfloor = 6 = \lfloor x \rfloor + 1.
$$
