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Math Help - Elementary Analysis Proof (inequalities)

  1. #1
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    [SOLVED] Elementary Analysis Proof (inequalities)

    Hi there,

    This is my first post on the forums, so I hope people can help me out. I'll be spending a lot of time here as I'm taking two difficult university math courses, Analysis I and Honours Linear Algebra.

    Anyways, here is the question, part of a suggested exercise for the Analysis class. I converted it to the form of a<b<c but that didn't help me. I'm new to the proofs and need a lot of help. Any help will be appreciated.


    If n \in \mathbb{N} prove

    <br />
 \left |1-{\frac {n}{n+1} \right| < \epsilon  \iff  n> \frac{1}{\epsilon} -1 \right<br /> <br />
    Last edited by manygrams; October 14th 2010 at 07:03 PM. Reason: solved
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  2. #2
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    <br />
 n> \frac{1}{\epsilon} -1 \iff n + 1 > \frac{1}{\epsilon} \iff \epsilon > \frac{1}{n + 1} = \frac{1 + n - n}{n + 1} = 1 - \frac{n}{n + 1} <br />

    Since n < n + 1 \Rightarrow \frac{n}{n+1} < 1

    So the absolute value bars are not really necessary here if n \ge 0
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  3. #3
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    That one is simple:

    \left| {1 - \frac{1}<br />
{{n + 1}}} \right| = \frac{1}<br />
{{n + 1}} < \varepsilon
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  4. #4
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    Thanks, I'm surprised I couldn't figure that one out!
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