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Let $X_n$ be a sequence of non negative real numbers. Then which of the following is true?

  1. $\liminf_{n\to\infty} X_n = 0 \implies \lim_{n\to\infty} X_n^2 = 0$
  2. $\limsup_{n\to\infty} X_n = 0 \implies \lim_{n\to\infty} X_n^2 = 0$
  3. $\liminf_{n\to\infty} X_n = 0 \implies X_n$ is bounded
  4. $\liminf_{n\to\infty} X_n^2 > 4 \implies \limsup_{n\to\infty} X_n > 4$

Please Help me to Prove or Disprove the given options.

My Attempt:

Option 1 can be discarded by taking the sequence (0,1,0,1,0,1,0,1,...)

Option 3 can be discarded by taking the sequence (0,1,0,2,0,3,0,4,0,5,0,6,...)

Option 4 can be discarded by taking the sequence (3,3,3,3,3,...)

Please help me to Prove option 2.

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    $\begingroup$ Have you tried anything? It'd be easier and better for you if you show what you tried and we help you with what you are having trouble. $\endgroup$ – ABP Jan 3 '20 at 2:22
  • $\begingroup$ Think about what $\limsup_{n\to\infty}x_n=0$ means for a sequence of non-negative numbers. $\endgroup$ – bjorn93 Jan 3 '20 at 2:23
  • $\begingroup$ Please help me how to prove option 2. $\endgroup$ – Baljeet Jan 3 '20 at 3:13
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If $\limsup_{n\to\infty}x_n=0$, then there exists $N$ such that $\sup_{n\geqslant N}x_n<1$, and hence $x_n^2<x_n$ for $n\geqslant N$. It follows then $$\limsup_{n\to\infty} x_n^2\leqslant \limsup_{n\to\infty} x_n=0,$$ and from $x_n\geqslant 0$ that $\liminf_{n\to\infty} x_n^2\geqslant 0$. Since trivially $\liminf_{n\to\infty} x_n^2\leqslant\limsup_{n\to\infty} x_n^2$, we find that $$ \liminf_{n\to\infty}x_n^2 = \limsup_{n\to\infty} x_n^2 = 0, $$ and hence $\lim_{n\to\infty} x_n^2=0$.

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  • $\begingroup$ Thanks for the help. $\endgroup$ – Baljeet Jan 3 '20 at 5:38
  • $\begingroup$ what if we take the sequence {0,-1.0,-1,0,-1,_ _ _ _ _ _}? $\endgroup$ – Mansi Aug 19 '20 at 2:55
  • $\begingroup$ @Huny Which of the four assertions are you proposing this as a counterexample? $\endgroup$ – Math1000 Aug 19 '20 at 15:32
  • $\begingroup$ Sorry, didn't notice non-negative real numbers!! Thanks for replying. :) $\endgroup$ – Mansi Aug 20 '20 at 11:54

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