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Math Help - Monotone Sequences

  1. #1
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    Question Monotone Sequences

    I could really use some help in one of my questions


    Let alpha be a fixed real number such that alpha is strictly less than 1/3. The sequence is defined recusively as follows:

    a1 = alpha an+1=1/4(1+an) for n greater or equal to 1


    (a) Show that an is less than or equal to 1/3for all n. (Use induction on n.)
    (b) Show that (an) is increasing
    (c) Deduce that (an) has a limit and determine lim(an) as n->infinity
    (d) Show that there exists an nsub0 greater or equal to 1 such that an>0 for all n greater or equal to nsub0

    I have done up to (c) but i can't find any notes on part (d). Can anyone help?

    Thanks
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  2. #2
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    Well, that is basically, the definition of limit isn't it? If lim a_n= L, then given any \epsilon> 0, there exist N such that if n> N, |a_n- L|< \epsilon. Presumably you got a positive number for the limt in (c) (otherewise (d) is not true!). Use the definition of limit with \epsilon equal to that limit.
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  3. #3
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    Quote Originally Posted by HallsofIvy View Post
    Well, that is basically, the definition of limit isn't it? If lim a_n= L, then given any \epsilon> 0, there exist N such that if n> N, |a_n- L|< \epsilon. Presumably you got a positive number for the limt in (c) (otherewise (d) is not true!). Use the definition of limit with \color{red}{\epsilon} equal to that limit.
    I think you meant to say L?
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