I think this goes in this section.

Let x be a real number. Prove that

abs(sin nx) <= n*abs(sinx) for all positive integers n.

Printable View

- Jul 5th 2006, 05:02 PMNichelle14Proof
I think this goes in this section.

Let x be a real number. Prove that

abs(sin nx) <= n*abs(sinx) for all positive integers n. - Jul 5th 2006, 05:41 PMThePerfectHackerQuote:

Originally Posted by**Nichelle14**

Thus, you need to prove that,

If and only if,

- Jul 8th 2006, 06:25 PMNichelle14
I still don't understand what to do.

- Jul 8th 2006, 08:42 PMThePerfectHacker
Mathematical induction is the tool here.

It is true for thus, there is a such as,

Thus,

But,

Thus, (mutiply inequalities notice they are non-negative),

......(1)

Also,

And,

Thus, (mutiply inequalites notice they are non-negative),

......(2)

Using the property on (1) and (2) we have,

Now, add these inequalities,

Now, by triangular inequality,

By transitivity ( ),

Thus, recognizing the sum for sine and simply the right side,

Proof is complete. - Jul 9th 2006, 04:58 AMQuick
Is it possible to say

(because it seems somewhat ridiculous) - Jul 9th 2006, 06:47 AMThePerfectHackerQuote:

Originally Posted by**Quick**

is defined to be true whenever,

OR .

Furthermore, in mathematics this expression is used a lot. There is one powerful theorem in set theory (Zorn's Lemma) which is based on the fact that .