Proof, Math Induction

1. Aug 29, 2010

phillyolly

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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2. Aug 29, 2010

epenguin

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.

3. Aug 29, 2010

phillyolly

4. Aug 29, 2010

jgens

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?

5. Aug 29, 2010

epenguin

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?

6. Aug 29, 2010

╔(σ_σ)╝

You are not doing it correctly.

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

You showed that
$$a_{2}= \frac{1}{2} + 1= 1.5$$ Correct ?

And you concluded that
$$1 \leq a_{2} \leq 2$$.

What you are not doing correctly is this same process for $$a_{k+1}$$

You have to show that
$$1 \leq a_{k+1} \leq 2$$

We know by inductive hypothesis that

$$1 \leq a_{k} \leq 2$$

From this we can tell that

$$\frac{1}{2} \leq \frac{a_{k}}{2} \leq 1$$

So what can you say about $$\frac{1}{a_{k}}$$ ?

What inequality does it satisfy, if we know that $$1 \leq a_{k} \leq 2$$ ?

After that, what can we say about the question marks below.
$$? \leq\frac{a_{k}}{2}+ \frac{1}{a_{k}} \leq ?$$

7. Aug 29, 2010

phillyolly

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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8. Aug 29, 2010

╔(σ_σ)╝

You have the right idea but you made a mistake in the inequality for $$\frac{1}{a_{n}}$$. Read your inequality and see if it makes sense.

9. Aug 30, 2010

epenguin

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.

10. Aug 30, 2010

╔(σ_σ)╝

Yes, i agree with the above post. You should seriously consider re-learning induction .

11. Aug 30, 2010

phillyolly

I see my mistake...

12. Aug 30, 2010

phillyolly

This is so new to me. I pass.

13. Aug 30, 2010