Define the sequence $x_1=1$ and for all $n\geq 1$ $x_{n+1}=1+\frac{n}{x_n}$
Does it follow that $x_n$ is an increasing sequence?
This is not a homework, I found the exercise to prove $\sqrt{n}\leq x_n \leq \sqrt{n}+1$ on a textbook. By curiosity, I was playing around with my computer to see the behavior of this sequence, which seems to increase very slow.
Further more, the condition $x_{n+1}\geq x_{n}$ would imply the stronger relation than given in the book, which is $x_n \leq \frac{1+\sqrt{1+4n}}{2}$.
Using this, one can show that for $k>2$ $\lfloor x_n \rfloor = k$ is satisfied for exactly $2k$ times values of $n$.