how do we get from here.

to here?

It's done in one step in my textbook, so I'm assuming it's obvious, but I can't see it.

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- March 6th 2013, 06:12 AMAlyoshaKazAnalysis inequality.
how do we get from here.

to here?

It's done in one step in my textbook, so I'm assuming it's obvious, but I can't see it. - March 6th 2013, 06:32 AMPlatoRe: Analysis inequality.
- March 6th 2013, 06:53 AMAlyoshaKazRe: Analysis inequality.
Ok, sorry. It's a proof for the Theorem:

"For each positive real number a and each integer n > 1, there is a unique real number b such that

In the special case

The proof begins with

etc.

to give decimal b^2 = (1.414...)^2 = 2

Then the method uses a table

b b b^2 1 1 1 1.4 1.4 1.96 1.41 1.41 1.9881 1/414 1/414 1.999396 etc.

to prove so.

Then to prove the least upper bound of the set of numbesr in the third column of the table is 2, we check that M=2 is an upper bound of E, which follows from the inequalities at the top. Then to show if M' < 2 there is a number in E that is greater then M'.

We put

Then we have

so

- March 6th 2013, 07:20 AMAlyoshaKazRe: Analysis inequality.
Also, on a side not, I'm not sure how the initial numbers are chosen for the inequalities at the beginning.

- March 7th 2013, 12:07 AMAlyoshaKazRe: Analysis inequality.
Is bumping allowed?