What is the boundedness theorem?

1 Answer
George C.
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)# is defined and continuous on a closed interval #[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.

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