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Math Help - Proof of sequence

  1. #1
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    Proof of sequence

    I was going over my practice exam for tonights final, and came across the following problem:

    Show that the sequence defined by
    a1 = 1
    and
    an+1 = 3 - 1/an

    is increasing and an < 3 for all n. Deduce that {an} is convergent and find its limit.

    Any time i try to prove the statement i end up assuming that the sequence is increasing in the actual proof.
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  2. #2
    o_O
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    Do you mean: a_{1} = 1 \qquad a_{{\color{red}n+1}} = 3 - \frac{1}{a_{n}}

    Remember that: Every monotonic, bounded sequence is convergent.

    Use induction to show that \forall n \geq 2 \ : \ 2 \leq a_{n} \leq 3 (i.e. it's bounded).

    And you can use induction again to show that a_{n+1} > a_{n} is true for all n (i.e. it's monotonic).

    Now this proves convergence. So, let \lim_{n \to \infty} a_{n} = L (we can assume this because we know a_{n} converges to some limit which we call L).

    Now take the limit from both sides of the recursive sequence:

    \begin{aligned}\lim_{n \to \infty} a_{n+1} & = \lim_{n \to \infty} \left(3 - \frac{1}{a_{n}}\right) \\ & \ \ \vdots \\ L & = 3 - \frac{1}{L} \end{aligned}

    and solve the quadratic, picking the appropriate solution.
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