Proof and Math Induction Homework | Step-by-Step Solution

[phillyolly]

Homework Statement

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

Homework Equations

The Attempt at a Solution

 

[epenguin]

It does not seem to me you are getting it correctly. If you are getting involved with a_k+2 as well as a_k+1 and a_k 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.

 

[phillyolly]

Please expand your suggestion...

 

[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?

 

[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?

 

[╔(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 ?

 

[phillyolly]

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.

 

[╔(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.

 

[epenguin]

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

In your second you leave out the statement about a_k+1 and make one about only a_k 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.

 

[╔(σ_σ)╝]

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

 

[phillyolly]

I see my mistake...

 

[phillyolly]

This is so new to me. I pass.

 

[╔(σ_σ)╝]

Can you re- read the chapter on induction in your txtbook and then ask specific questions about what you do not understand.