Re: Least Upper Bound Proof

Re: Least Upper Bound Proof

Quote:

Originally Posted by

**Plato** Let

. Suppose that

. (We know that

WHY?)

since i know pL is a upper bound of pA the least upper bound of pA must be less than or equal to pL.

Quote:

Originally Posted by

**Plato** It follows that

so

WHY?

since i know L is the least upper bound of A, then

Quote:

Originally Posted by

**Plato** How is that a contradiction?

not to sure on this one. is it because is supposed to equal L but by the proof is less than L?

Re: Least Upper Bound Proof

Quote:

Originally Posted by

**JBrandon** not to sure on this one. is it because

is supposed to equal L but by the proof is less than L?

Well that implies that . Can that be?

Re: Least Upper Bound Proof

so the full proof should be.

Let , , , , and p be a positive number.

Since , and , then .

Since , then .

therefor .

Assume

since , and , then .

since , then

since , and , then

since , then . a contradiction as must be true for

therefor and

Re: Least Upper Bound Proof

Quote:

Originally Posted by

**JBrandon** so the full proof should be.

Let

,

,

,

, and p be a positive number.

Since

, and

, then

.

Since

, then

.

therefor

.

Assume

since

, and

, then

.

since

, then

since

, and

, then

since

, then

. a contradiction as

must be true for

therefor

and

Honestly, I don't know how to answer you questions.

I taught analysis for more than 35 years.

I drilled into students that if then for any then it must be the case that .

Oddly enough, that turns out to be the most important idea in analysis.

So if because that must be a contradiction because