Prove a sequence is nonincreasing by induction

I'm stuck on this....

Problem:

Show that (tn) is a nonincreasing sequence. That is, use induction to prove that where

I have the base case so, . Then assume .

But I'm stuck at this part. I don't know how to arrive at . I tried messing with the inequalities and stuff, but I'm still not sure.

Edit: The only thing I could really come up with was .....

then use the fact and so

Re: Prove a sequence is nonincreasing by induction

Quote:

Originally Posted by

**JSB1917** Problem:

Show that (tn) is a nonincreasing sequence. That is, use induction to prove that

where

You better double check the problem!

**is an increasing sequence.**

Re: Prove a sequence is nonincreasing by induction

Yeah, sorry, it's actually I forgot the at the end. I put that in the original post now.

Re: Prove a sequence is nonincreasing by induction

Quote:

Originally Posted by

**JSB1917** Yeah, sorry, it's actually

I forgot the

at the end. I put that in the original post now.

Suppose you know that .

Then multiply:

But

So

Re: Prove a sequence is nonincreasing by induction

Quote:

Originally Posted by

**Plato** Suppose you know that

.

Then multiply:

But

So

Yeah, but I don't think it's correct that

Because....

, then

and so

Re: Prove a sequence is nonincreasing by induction

Quote:

Originally Posted by

**JSB1917** Yeah, but I don't think it's correct that

Because....

, then

and so

You are correct about that. Sorry, I got lost in my own notation.

I still have your OP in my head. It is odd how the brains holds on to things.

Re: Prove a sequence is nonincreasing by induction

Quote:

Originally Posted by

**Plato** You are correct about that. Sorry, I got lost in my own notation.

I still have your OP in my head.

I know that we need to work backwards from

.

I think using that then

and so,

and so

That seems right to me. Thoughts?

Re: Prove a sequence is nonincreasing by induction

Quote:

Originally Posted by

**JSB1917** and so Thoughts?

I do not follow that. As I said above, the brain just get stuck on an idea and will not let it go.

Think about this. We know .

Consider .

What does that tell us?

Re: Prove a sequence is nonincreasing by induction

Quote:

Originally Posted by

**Plato** I do not follow that. As I said above, the brain just get stuck on an idea and will not let it go.

Think about this. We know

.

Consider

.

What does that tell us?

Aside from it's negative, not sure.

Re: Prove a sequence is nonincreasing by induction

Quote:

Originally Posted by

**JSB1917** Aside from it's negative, not sure.

Surely says

Re: Prove a sequence is nonincreasing by induction

Quote:

Originally Posted by

**Plato** Surely

says

yeah...I know that, but where is this leading? I'm just not seeing it.

Re: Prove a sequence is nonincreasing by induction

If we have:

then since , and therefore the sequence is decreasing.

- Hollywood