Let {E_n} be a sequence of subsets of \(\mathbb{R}\).
Define
\(\limsup _n E_n=\bigcap_{k=1}^{\infty} \bigcup_{n=k}^{\infty} E_n\)

\(\liminf _n E_n=\bigcup_{k=1}^{\infty} \bigcap_{n=k}^{\infty} E_n\)

Which of the following statements is true?

Concept -

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

Calculation:

Let {En} be a sequence of subsets of R 

\(\limsup _n E_n=\bigcap_{k=1}^{\infty} \bigcup_{n=k}^{\infty} E_n\)  and 

\(\liminf _n E_n=\bigcup_{k=1}^{\infty} \bigcap_{n=k}^{\infty} E_n\)

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

option 1 is incorrect

x \(∈\) \(\ {\cap}\) A_i imply x ∈ A_i 

x \(∈\)\(\ {\cap}\)(\(\ {\cup}\)E_n )

x \(∈\)\(\ {\cup}\) E_n ( finite )

Hence option (2) & (4) are incorrect 

Hence option (3) is correct