# Checking the nature of the sequence ${x_{n}}=\frac{\left\lfloor xn \right\rfloor}{n}$ **[TIFR GS-2011]**

[TIFR GS-2011, Q1.]

Consider the sequence ${x_{n}}$ defined by $\frac{\left\lfloor xn \right\rfloor}{n}$ for $x \in \mathbb{R}$. Then

(A) Converges to $x$.

(B) Converges not to $x$.

(C) doesn't converge.

(D) oscillate.

When I take $x=1$ sequence converges to $1$. similarly I had given particular values to $x$, I am not able to judge the answer. Can any one help me to figure out the solution? Thanks in advance.

• $\frac{\lfloor xn \rfloor}{n}=\frac{xn}{n}-\frac{\{xn\}}{n}$ – A.Γ. Oct 9 '17 at 14:47
• Note that $\lfloor{xn}\rfloor$ is within $1$ of $xn$. – anomaly Oct 9 '17 at 14:53

$$x_n=\frac{\lfloor xn\rfloor}{n}=\frac{xn-\{xn\}}{n}=x-\frac{\{xn\}}{n}\to x\text{ as }n\to\infty,$$ where $\{y\}\in[0,1)$ is the fractional part of $y$. Therefore, the answer is (A).