1. D

    Arithmetics of superior limits

    I uploaded the questions that I have, thank you in advance.
  2. M

    Limit Superior Definition Proof

    Maybe someone can help me out, I thought I had an idea but I don't think it is right. I am trying to prove the above. My attempt: I am given that the lim sup Xn is finite, which I think means it exists, which I will call L. So I have a sequence [Xn] = [x1, x2, x3,... xn, xn+1,....] and...
  3. J

    Limit superior and limit inferior

    I'm stock with this: I don't understand quite well. Can anyone help me? I'd appreciate any help.
  4. E

    Limit superior proof

    Hi there, I'm working through a basic proof in a probability theory book, and I'm stuck trying to work out whether one of the statements is a typo, or I'm missing something: A_k is an arbitrary indexed set If...
  5. A

    limit superior of sequence

    Let (x_n) be a bounded sequence. For each n \in \mathbb{N} , let y_n=x_{2n} and z_n=x_{2n-1}. Prove that \limsup{x_n}=\max({\limsup{y_n},\limsup{z_n}}). I tried using the identity \max(x,y)=\frac{1}{2}{(x+y-|x-y|)}, but it can't seem to work... can anyone help me out?
  6. A

    Limit Superior and Limit Inferior

    I was just wondering if my method for finding lim sup and lim inf for sequences is correct. It's different to how we do it in class, but I asked my lecturer about my way and he said that it looks fine but he'd think about it and get back to me. Essentially what I do is first take the sequence to...
  7. C

    Divergence of limit superior

    \lim\sup x_n=-\infty\implies \lim x_n=-\infty, does the converse hold? How to prove this?
  8. C

    When limit superior diverges

    Prove that \lim\sup x_n=\infty\iff \forall M>0,\forall n\in\mathbb N,\exists k_0\ge n so that x_k>M. I think is an easy problem, but I'm confused, the statement establishes that x_k is bounded below, but not above. How to prove this?
  9. C

    Limit superior and limit inferior

    I get the definition of limit superior and limit inferior, but I can't start a proof for these properties: \underline{\lim }{\,{x}_{n}}=-\overline{\lim }\left(-{{x}_{n}}\right) \underline{\lim }{\,{x}_{n}}\le\overline{\lim }{\,{x}_{n}} \underline{\lim }{\,{x}_{n}}+\underline{\lim...
  10. A

    limit superior inequality

    Let an, bn be two bounded sequences. Show that limn-->∞ sup (an+bn) ≤ limn-->∞ sup an + limn-->∞ sup bn
  11. N

    limits superior and inferior

    suppose \lim sup_{n\to\infty}A_n=\bigcap_{n=1}^{\infty}\bigcup_{m\ge n}A_m and \lim inf_{n\to\infty}A_n=\bigcup_{n=1}^{\infty}\bigcap_{m\ge n}A_m where \bigcup_{m\ge n}A_m=A_n\cup A_{n+1}\cup .... and \bigcap_{m\ge n}A_m=A_n\cap A_{n+1}\cap .... Then given that \lim...
  12. F

    limit superior ....

    Find \overline{lim} \ a_n where : a_n = \begin{cases} 1 &, \text{ if } n\ is \ square \\ 0 &, \text{ if } n \ not \ square \end{cases}
  13. M

    Limit Superior

    Suppose that \displaystyle{x_n} is a bounded sequence of real numbers.Show that \displaystyle{y_n=\sup_{m \geq n}x_m} is a bounded decreasing sequence and deduce that \displaystyle{y_n \longrightarrow y},ie \displaystyle{\limsup_{n\to\infty}x_n=y}.
  14. S

    limit superior and limit inferior

    I dont understand limit superior and limit inferior. I know what the definitions are but I would not know how to apply it in an example since my teacher did not explain how to go about that. How would I find the limit superior and limit inferior of {(n/3)-[n/3]}?
  15. I

    sequence - limit superior

    I need to prove the following: Let Sn be a sequence in R. If s in R and for every e > 0, there exists n' in N s.t. Sn < s + e for all n >= n', prove that Lim superior Sn <= s Here is how I am approaching it: first, Sn - s < e -> s is the limit of Sn as n-> infinity. I also know that Lim...
  16. K

    Limits and Limit Superior proof

    By definition, a_n->a iff for all ε>0, there exists an integer N such that n≥N => |a_n - a|< ε [note: also under discussion in Math Links forum]
  17. K

    Real analysis: Limit superior proof

    [note: also under discussion in s.o.s. math board]
  18. T

    Limit, Limit Superior, and Limit Inferior of a function

    Prove that \lim _{x \rightarrow a } f(x) = \alpha if and only if \limsup _{x \rightarrow a} f(x) = \liminf _{x \rightarrow a} f(x) = \alpha . Proof so far: Suppose that \limsup _{x \rightarrow a} f(x) = \liminf _{x \rightarrow a} f(x) = \alpha , then for \delta > 0 , we have: \inf...
  19. W

    Find the superior and inferior limit??

    Hi guys: the Qs is 'Find the superior and inferior limit of the sequence"(-1)^n(1+n^-1)". Even I know the answer for the Qs is 1 and -1, but what should be the currect process for this Qs, please give some detail.thanks!! <img smilieid="84" class="inlineimg"...
  20. L

    Which is the superior method of teaching?

    I am not a maths teacher so have no idea how to work this out. I used two methods of teaching (A and B) to teach my 70 students. I then asked the students 50 multiple choice questions on material taught with each method; each question had 5 possible responses. The total number of correct...