Proving The Theorems
We look at the proof for the upper bound and the maximum of f. By applying these results to the function –f, the existence of the lower bound and the result for the minimum of f follows. Also note that everything in the proof is done within the context of the real numbers.
We first prove the boundedness theorem, which is a step in the proof of the extreme value theorem. The basic steps involved in the proof of the extreme value theorem are:
- Prove the boundedness theorem.
- Find a sequence so that its image converges to the supremum of f.
- Show that there exists a subsequence that converges to a point in the domain.
- Use continuity to show that the image of the subsequence converges to the supremum.
Read more about this topic: Extreme Value Theorem
Famous quotes containing the word proving:
“The momentary charge at Balaklava, in obedience to a blundering command, proving what a perfect machine the soldier is, has, properly enough, been celebrated by a poet laureate; but the steady, and for the most part successful, charge of this man, for some years, against the legions of Slavery, in obedience to an infinitely higher command, is as much more memorable than that as an intelligent and conscientious man is superior to a machine. Do you think that that will go unsung?”
—Henry David Thoreau (1817–1862)