What is the boundedness theorem?

1 Answer
Oct 14, 2017

A continuous function defined on a closed interval has an upper (and lower) bound.

Explanation:

Probably the simplest boundedness theorem states that a continuous function defined on a closed interval has an upper (and lower) bound.

Proof by contradiction

Suppose f(x)f(x) is defined and continuous on a closed interval [a, b][a,b], but has no upper bound.

Then:

AA n in NN, EE x_n in [a, b] : f(x_n) > n

Since the sequence of x_n's lies in a bounded interval, it is dense at some point in the closure of the interval. Since the interval is closed, that must be at some point c actually in the interval [a, b].

Since the sequence of x_n's is dense at c, there is some monotonically increasing sequence n_k in NN such that x_(n_k) -> c as k->oo.

Now f(x) is continuous at c, so:

lim_(x->c) f(x) = f(c)

which is bounded.

But:

lim_(k->oo) x_(n_k) = c" " and " "lim_(k->oo) f(x_(n_k)) = oo

is unbounded.

...contradiction.

So there is no such f(x) lacking upper (or lower) bound.