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Proof, Math Induction

  1. Aug 29, 2010 #1
    1. The problem statement, all variables and given/known data

    Just wanted to check if I am getting it correctly before I proceed further. Thank you!

    2. Relevant equations



    3. The attempt at a solution
     

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  3. Aug 29, 2010 #2

    epenguin

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    It does not seem to me you are getting it correctly. If you are getting involved with ak+2 as well as ak+1 and ak I don't think you are going to get it.

    I suggest you write out "The statement will be proved for n=(k+1) of we can prove that ...". That will guide your further steps.
     
  4. Aug 29, 2010 #3
    Please expand your suggestion...
     
  5. Aug 29, 2010 #4

    jgens

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    Why don't you try epenguin's suggestion first? Aside from the fact, based on the number of induction threads you've posted here today, you're clearly not comfortable with the method. I found that once I really understood the method intuitively, I became a lot more confident with my ability to write proofs by induction. So, is there anything that you find particularly discomforting or unclear about induction?
     
  6. Aug 29, 2010 #5

    epenguin

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    The next step is the whole idea of induction. Consult any examples you have done or followed.

    In an induction proof a statement about a formula F(n) is true for n=k. What do you have to prove in order to be able to argue it is true for all n greater than k?
     
  7. Aug 29, 2010 #6
    You are not doing it correctly.

    When you showed that this [tex] 1 \leq a_{n} \leq 2[/tex] inequality holds when n=2, how did you do it ?

    You showed that
    [tex] a_{2}= \frac{1}{2} + 1= 1.5[/tex] Correct ?

    And you concluded that
    [tex] 1 \leq a_{2} \leq 2[/tex].

    What you are not doing correctly is this same process for [tex] a_{k+1}[/tex]

    You have to show that
    [tex]1 \leq a_{k+1} \leq 2 [/tex]

    We know by inductive hypothesis that

    [tex]1 \leq a_{k} \leq 2 [/tex]

    From this we can tell that

    [tex]\frac{1}{2} \leq \frac{a_{k}}{2} \leq 1 [/tex]

    So what can you say about [tex]\frac{1}{a_{k}}[/tex] ?

    What inequality does it satisfy, if we know that [tex]1 \leq a_{k} \leq 2 [/tex] ?


    After that, what can we say about the question marks below.
    [tex]? \leq\frac{a_{k}}{2}+ \frac{1}{a_{k}} \leq ?[/tex]
     
  8. Aug 29, 2010 #7
    Thank you for your critical and helpful comments, PH residents.
    I know, I have annoyed all of you with my posts.
    But thanks to you, I have learned a lot in the past couple of days (considering my previous zero exposure to Real Analysis).

    Very thankful.
     

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  9. Aug 29, 2010 #8
    You have the right idea but you made a mistake in the inequality for [tex] \frac{1}{a_{n}}[/tex]. Read your inequality and see if it makes sense.
     
  10. Aug 30, 2010 #9

    epenguin

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    In your first attempt you made a statement about ak+1 and ak, and you then went on to make another about ak+1 and ak and ak+2 which will get you nowhere.

    In your second you leave out the statement about ak+1 and make one about only ak which will equally get you nowhere by itself. Try a happy medium! Better, please look up some other elementary example of an induction proof, because you do not (yet) have a problem with real analysis, you have a problem with induction. Which is a very easy idea - it might have applications or examples in real analysis which are not so easy, but you must get the easy part clear.
     
  11. Aug 30, 2010 #10
    Yes, i agree with the above post. You should seriously consider re-learning induction .
     
  12. Aug 30, 2010 #11
    I see my mistake...
     
  13. Aug 30, 2010 #12
    This is so new to me. I pass.
     
  14. Aug 30, 2010 #13
    Can you re- read the chapter on induction in your txtbook and then ask specific questions about what you do not understand.
     
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