Math Is Fun Forum

sumpm1

Member

Registered: 2007-03-05

Posts: 42

Real Analysis Help

Hey guys, need some more help here. We have been covering absolute values and the Triangle inequality theorem in class.

Triangle Inequality Theorem:

Reverse Triangle Inequality:

1. Let

and

be two nonempty sets or real numbers, and let

be a real number. Suppose that there exist positive real numbers

and

such that

for all

and that

for all

. Use the Triangle Inequality Theorem to prove that

for all

and

.

2. Let

be a nonempty set of real numbers, and let

be a nonzero real number. Suppose that

for all

. Prove that

for all

.

3. Let

and

be real numbers. Suppose that

for all positive numbers

. Prove that

.

In number 1, I thought maybe I could add multiples of the inequalities

and

. But I am not sure how I should use the Triangle Inequality to get to

.

In number 2, It seems like maybe I should be using the arithmetic mean

somewhere. And this is in this section of the book, but we have not covered this yet, and I doubt the professor would assign a problem that needed this without presenting it in class. So the  key is probably again the Triangle Inequality or the Reverse Triangle Inequality.

And problem 3 I don't have a clue on.

I would appreciate some hints on this if anyone can provide some insight. I feel like this is less about UNDERSTANDING the concepts, and more about RECOGNIZING TRICKS!!! Perhaps they are one in the same, but I always have trouble finding the "trick."

Thanks

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Real Analysis Help

#1.

And finally

; do the same as above, then put all the results together.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Real Analysis Help

#2.
Use the first half of the reverse triangle inequality.

#3.
Suppose to the contrary that

.

Let

.

Then

from which a contradiction immediately follows.

sumpm1

Member

Registered: 2007-03-05

Posts: 42

Re: Real Analysis Help

Thank you so much Jane.

In #2, I had to take

Is this all correct?

And in #3 you found that

I am having trouble arriving at this and am finding instead that

, but I am not confident in my understanding of how to manipulate inequalities in this way.

For an example of my understanding of manipulating inequalities: We let

. So I would follow that

and then that

, is this correct? Also, It may seem kind of DINGBAT of me... but I am getting hung up when moving

to the right side of the inequality and leaving the space empty:

.

Also this is troublesome to me and I need a better understanding of how to manipulate inequalities. Or perhaps it is the misuse of an EQUALITY and an INEQUALITY together as in the following manipulation(more DINGBAT):

Am I going about this all wrong?

Thanks

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Real Analysis Help

Is this all correct?

Yes, that is correct.

And in #3 you found that

What I mean is

Sorry if I didn’t make it clear.

sumpm1

Member

Registered: 2007-03-05

Posts: 42

Re: Real Analysis Help

Ahhh... My fault, I was thinking that letting

, but you were simply restating the given in the problem!!!

Thanks again.