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मान लीजिए {En} \(\mathbb{R}\)  के उपसमुच्चयों का अनुक्रम है।

मानें कि

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

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

निम्न कथनों में से कौन-सा सत्य है?

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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 कुछ n के लिए}
  3. liminfn En = {x ∶ x ∈ En सभी के लिए लेकिन परिमित रूप से अनेक n के लिए}
  4. liminfn En = {x ∶ x ∈ En अनंत रूप से अनेक n के लिए}

Answer (Detailed Solution Below)

Option 3 : liminfn En = {x ∶ x ∈ En सभी के लिए लेकिन परिमित रूप से अनेक n के लिए}
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Detailed Solution

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अवधारणा-

(i) यदि अनुक्रम xn अभिसारी है, तो limsupn En = liminfn En

गणना:

मान लीजिये {En} R का एक उपसमुच्चयों का अनुक्रम है

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

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

विकल्प 1 के लिए, यदि अभिसारी है तो limsupn En = liminfn En

विकल्प 1 गलत है।

x \(∈\) \(\ {\cap}\) Ai का अर्थ है x ∈ Ai

x \(∈\)\(\ {\cap}\)(\(\ {\cup}\)En )

x \(∈\)\(\ {\cup}\) En ( परिमित )

इसलिए विकल्प (2) और (4) गलत हैं।

इसलिए विकल्प (3) सही है।

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