boundedness theorem
Boundedness Theorem.
Let and be real numbers with , and let be a continuous, real valued function on . Then is bounded above and below on .
Proof.
Suppose not. Then for all natural numbers we can find some such that . The sequence
is bounded
, so by the Bolzano-Weierstrass theorem
it has a convergent
sub sequence, say . As is closed converges
to a value in . By the continuity of we should have that converges, but by construction it diverges. This contradiction
finishes the proof.
Title | boundedness theorem |
---|---|
Canonical name | BoundednessTheorem |
Date of creation | 2013-03-22 14:29:18 |
Last modified on | 2013-03-22 14:29:18 |
Owner | classicleft (5752) |
Last modified by | classicleft (5752) |
Numerical id | 6 |
Author | classicleft (5752) |
Entry type | Theorem |
Classification | msc 26A06 |