What is the boundedness theorem?
1 Answer
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
Then:
#AA n in NN, EE x_n in [a, b] : f(x_n) > n#
Since the sequence of
Since the sequence of
Now
#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