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{En} को \(\mathbb{R}\) के उपसमुच्चयों का अनुक्रम मानें।

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 परिमितानेक n के अतिरिक्त सब के लिए}
  4. liminfn En = {x ∶ x ∈ En  अनंत: बहु n के लिए}

Answer (Detailed Solution Below)

Option 3 : liminfn En = {x ∶ x ∈ En परिमितानेक n के अतिरिक्त सब के लिए}
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Q1.Let \(\rm S =\left\{x \in R:x >1 \ and \ \frac{1-x^4}{1-x^3}>22\right\}\) Which of the following is true about S? 
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Q6.{En} को \(\mathbb{R}\) के उपसमुच्चयों का अनुक्रम मानें। 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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