Unizor - Function Limit - Bounded Functions

Unizor - Creative Minds through Art of Mathematics - Math4Teens

Bounded Functions

In this lecture we will consider real functions f(x) of real argument x.
The domain of these functions will be a contiguous interval, finite or infinite, including or not including the endpoints.
A finite contiguous interval with endpoints included we will call segment.

We will prove the following theorem.

Boundedness Theorem

A continuous function defined on a segment (finite interval with endpoints) is bounded from above and from below.

Proof

The proof is based on two main properties introduced in prior lectures:
(a) Bolzano - Weierstrass Theorem that states that from any bounded sequence we can extract a convergent subsequence.
(b) Continuity property.

We will only prove the boundedness from above, the one from below is completely analogous.

Let's assume the opposite, that our function f(x) is defined and continuous on segment [a,b], and is not bounded from above.
Then for any, however large, number N we will be able to find an argument xN∈[a,b] such that f(xN) ≥ N.

The sequence {xN} consists from points in segment [a,b] and, therefore, is bounded by the endpoints of this segment.
According to Bolzano - Weierstrass Theorem, we can extract from it a subsequence {yn} of points in this segment that converges to some point Y∈[a,b].
Since a set of values of our function is unbounded on sequence {xN}, it is also unbounded on subsequence {yn} and, therefore, there no limit of f(yn) as n→∞.

But function f(x) is continuous on segment [a,b], which means that, if {yn}→Y∈[a,b], then {f(yn)}→f(Y). So, the limit of f(yn) does exist.
Came to a contradiction. Hence, f(x) is bounded from above.

[ZorShekhtman]

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