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TIFR-2011-Maths-A-1

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Consider the sequence $\left \{x_{n}  \right \}$ defined by $x_{n}=\frac{\left[nx\right]}{n}$ for $x \in \mathbb{R}$ where $[·]$ denotes the integer part. Then $\left \{x_{n}  \right \}$

  1. Converges to $x.$
  2. Converges but not to $x.$
  3. Does not converge 
  4. Oscillates 
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We know that $n<x<n+1$Where $n\in \mathbb{N}$ then $[x]=n$So $nx-1< [nx]\le nx \Rightarrow x-\frac{1}{n}<\frac{[x]}{n}\le x$ Now by Sandwitch therem limit is $x$
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