How can i prove that $ \forall x \in \mathbb{R} \displaystyle \lim_{n \to \infty} \dfrac{\left \lfloor{x}\right \rfloor+\left \lfloor{2x}\right \rfloor+\cdots+\left \lfloor{nx}\right \rfloor}{n^2} = \dfrac{x}{2} $?
I tried $A_{n}=\dfrac{\left \lfloor{x}\right \rfloor+2\left \lfloor {x}\right \rfloor+\cdots+n \left \lfloor{x}\right \rfloor}{n^2}$, $B_{n}= \dfrac{\left \lfloor{x}\right \rfloor+\left \lfloor{2x}\right \rfloor+\cdots+\left \lfloor{nx}\right \rfloor}{n^2}$, $C_{n}= \dfrac{x+2x+\cdots +nx}{n^2}$
So $A_{n} \leq B_{n} \leq C_{n}$ $ \Rightarrow \displaystyle \lim_{n \to \infty}A_{n} \leq \displaystyle \lim_{n \to \infty} B_{n} \leq \displaystyle \lim_{n \to \infty} C_{n} \iff \dfrac{\left \lfloor{x}\right \rfloor}{2} \leq \displaystyle \lim_{n \to \infty} B_{n} \leq \dfrac{x}{2}$
But I dont know if thats all or Im missing something
It works if you replace your $ A_n $ with the following:
$$ A_n = \dfrac {(x-1) + (2x - 1) + ... + (nx - 1)} {n^2} = \dfrac {(x + 2x + ... + nx) - n} {n^2} = C_n - \dfrac {1} {n}$$
Then $$ \lim_{n \to \infty}A_n = \lim_{n \to \infty} C_n - \dfrac {1}{n} =\lim_{n \to \infty} C_n = \dfrac {x}{2} $$