Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!

share|cite|improve this question
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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.