This is a challenge question:

Let with and . If there exists a continuous function such that is non-decreasing, show there exists a such that .

Moderator edit:Approved Challenge question.

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- Aug 1st 2010, 06:42 AMDinkydoeSmall Challenge # 2
This is a challenge question:

Let with and . If there exists a continuous function such that is non-decreasing, show there exists a such that .

**Moderator edit:**Approved Challenge question. - Aug 1st 2010, 08:16 PMIondor
__Lemma 1:__

Let and .

There can be no sequence with for all n

and

no sequence with for all n

**Proof:**

Towards a contradiction suppose there is such a sequence with .

Since is non-decreasing and , we have

Since g is continous, , so that taking limits on both sides leads to the contradiction

.

Analogous in the second case.

qed

Now consider . , because is closed.

Assume .

case 1:

for some .

Then and for all according to the definition of y.

So is nonempty such that there is a sequence with , which violates lemma 1.

case 2:

for some

Then .

So is nonempty such that there is a sequence with and which violates lemma 1.

Therefor

.

qed - Aug 3rd 2010, 06:55 AMDinkydoe
- Aug 3rd 2010, 07:45 AMDinkydoe
You guys seem to have no trouble solving my Small challenges.... hehe (Rofl)

Here's what I did:

Let . Then and . We need to show the existence of a such that .

Let and .

1.Suppose .

Then is nonempty and by definition of we have for all .

Thus we must have by continuity of . But then . This contradicts the fact that is non-decreasing.

2. If we get the same contradiction: . Hence

__Conclusion__: