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Math Help - Difference between a sequence being bounded and having a limit?

  1. #1
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    Difference between a sequence being bounded and having a limit?

    Hey

    I am taking an analysis course at university. We covered things being bounded and then did limits. Going over my notes I realise I'm not entirely sure what the difference is... i.e. if a sequence has a limit (tending to infinity) of two, doesn't that make it bounded by two? Are the bound and the limit tending to infinity the same thing?

    Could someone please explain the differences between them?

    Thank you very much.
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  2. #2
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    Quote Originally Posted by Lisa1991 View Post
    Hey

    I am taking an analysis course at university. We covered things being bounded and then did limits. Going over my notes I realise I'm not entirely sure what the difference is... i.e. if a sequence has a limit (tending to infinity) of two, doesn't that make it bounded by two? Are the bound and the limit tending to infinity the same thing?

    Could someone please explain the differences between them?

    Thank you very much.
    Dear Lisa1991,

    First let me give you the definitions for "Limit of a sequence" and "Boundedness".

    Limit of a sequence.

    A sequence [a_{n}]_{n=1}^{\infty} is said to converge to a\in{R} if \forall~\epsilon>0~\exists~n_{0}\in{Z^{+}} such that whenever n>n_{0}\Rightarrow{\left|a_{n}-a\right|<\epsilon}

    Boundedness of a sequence.

    A sequence is said to be bounded if \exists~M\geq{0} such that, \left|a_{n}\right|\leq{M}~\forall~n\in{Z^{+}}

    Therefore boundedness and the limit tending to infinity is not the same. Anyhow it could be shown that every sequence which is convergent (has a limit) is bounded. I am also doing Real analysis in our university. Therefore if you have anything to discuss regarding this matter please don't hesitate to do so.

    Hope this helps.
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  3. #3
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    The sequence a_n= \left(-1\right)^n is bounded but has no limit.

    It is easy to show that if a sequence has a limit, then it must be bounded.

    If a sequence is bounded, then it does not necessarily have a limit but it must have subsequential limits- and you can show that every member of the sequence is a member of some convergent subsequence.
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    Quote Originally Posted by Lisa1991 View Post
    Hey

    I am taking an analysis course at university. We covered things being bounded and then did limits. Going over my notes I realise I'm not entirely sure what the difference is... i.e. if a sequence has a limit (tending to infinity) of two, doesn't that make it bounded by two? Are the bound and the limit tending to infinity the same thing?

    Could someone please explain the differences between them?

    Thank you very much.
    Only if the sequence is indreasing ,then two is un upper bound and actually the least upper bound.

    Generaly if the sequence has a limit two we have that:

    for all ε>0 there exists a natural No κ such that :

    for all ,n : n\geq\kappa\Longrightarrow |x_{n}-2|<\epsilon

    OR

    for all, n: n\geq\kappa\Longrightarrow |x_{n}|< 2+\epsilon.

    And if we choose now L as:

    L = max{ |x_{1}|,|x_{2}|,|x_{3}|.......|x_{\kappa}|,2+\epsi  lon},then:

    for all natural Nos, n : |x_{n}|\leq L,thus L is an upper bound

    Now if the sequence increases we have:

     x_{1}<x_{2}<x_{3}........x_{\kappa}<x_{\kappa+1}...........infinite dots <2+ε ,and as n goes to infinity ε goes to zero and the bound L BECOMES two
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    Thanks, everyone.
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