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f ∶ \(\mathbb{N} \rightarrow \mathbb{N}\) be a bounded function. Which of the following statements is NOT true?

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CSIR-UGC (NET) Mathematical Science: Held on (26 Nov 2020)
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  1. \(\limsup _{n \rightarrow \infty} f(n) \in \mathbb{N}\)
  2. \(\liminf _{n \rightarrow \infty} f(n) \in \mathbb{N}\)
  3. \(\liminf _{n \rightarrow \infty}(f(n)+n) \in \mathbb{N}\)
  4. \(\limsup _{n \rightarrow \infty}(f(n)+n) \notin \mathbb{N}\)

Answer (Detailed Solution Below)

Option 3 : \(\liminf _{n \rightarrow \infty}(f(n)+n) \in \mathbb{N}\)
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Calculation

Given 

f : \(\mathbb{N} \rightarrow \mathbb{N}\) be a bounded function .

let f(n) = 1 

taking constant function .

sup{f(n)} = 1 and inf{f(n)} = 1 

option 1 and 2 are correct . 

\(\lim_{x \to \infty } (1+n) \) = \(\ \infty\) 

does not belong to natural number .

option 4 is correct .

according to question 

hence correct option is 3 .

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