# Homework Help: 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

#### Attached Files:

• ###### pic.jpg
File size:
15.8 KB
Views:
139
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.

#### Attached Files:

• ###### pic.jpg
File size:
34.4 KB
Views:
103
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