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Let $(p_n)$ be a sequence with real numbers such that $\liminf (p_n)=-10$ and $\limsup (p_n)=10$. Now, are the following questions true or not true. (Prove answers). 1.)The sequence $(p_n)$ does not converge. 2.) That $(p_n)$ is both monotone increasing/decreasing. 3.) That $(p_n)$ is bounded.

1.)We know $(p_n)$ does not converge $\liminf\neq \limsup$.

2.)Help please!

3.) Since liminf and limsup $\neq$ to either $\infty$ or $-\infty$ we know $(p_n)$ is bounded.

Are my 1.) and 3.) right??? and can i get help on 2 please!

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1 and 3 are correct. For 2, Use monotone convergence thm – ILoveMath Oct 18 '12 at 6:42
up vote 1 down vote accepted

Part 2: Every monotone sequence has a limit by monotone convergence theorem. So your sequence $(p_n)$ cannot be monotone.

For part 3 I think you could add more details explaining why it is bounded. (But perhaps you have seen this as a theorem in your lecture?)

Anyway if $\limsup p_n=10$ then you know (from the definition of limit superior) that there exists $n_0$ such that $$n\ge n_0 \Rightarrow p_n\le 11.$$ This implies $$p_n \le \max \{11, p_1, p_2, \dots, p_{n_0}\}$$ for each $n$. So we have shown that $p_n$ is bounded from above.

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