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Let {En} be a sequence of subsets of .
Define

Which of the following statements is true?

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CSIR-UGC (NET) Mathematical Science: Held on (26 Nov 2020)
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  1. limsupn En = liminfn En 
  2. limsupn En = {x ∶ x ∈ En for some n}
  3. liminfn En = {x ∶ x ∈ En for all but finitely many n}
  4. liminfn En = {x ∶ x ∈ En  for infinitely many n}

Answer (Detailed Solution Below)

Option 3 : liminfn En = {x ∶ x ∈ En for all but finitely many n}
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Detailed Solution

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Concept -

(i) If the sequence xn convergent  then limsupn En = liminfn En 

Calculation:

Let {En} be a sequence of subsets of R 

  and 

for option 1, if convergent  then limsupn En = liminfn En 

option 1 is incorrect

x   Ai imply x ∈ Ai 

x (En )

x  En ( finite )

Hence option (2) & (4) are incorrect 

Hence option (3) is correct 

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