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.
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.
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
You guys seem to have no trouble solving my Small challenges.... hehe
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: