Monotone Convergence Theorem asserts that a monotone non-decreasing bounded sequence of real numbers has a limit. More accurately, a sequence a_n of real numbers which is
- monotone non-decreasing: \forall n\in N\quad a_n \le a_{n+1},
- bounded from above: \exists B \forall n\quad a_n \le B.
has a limit c = \lim_{n\to\infty} a_n .
The proof depends on the completeness? property of real numbers of which one incarnation is the existence of the greatest lower bound \inf of a non-empty set bounded from below. The set of all bounds B of the sequence a_n is such a set implying the existence of c such that c = inf \{B\}.
We wish to prove that, for the so defined c , c = \lim_{n\to\infty} a_n .
Let \epsilon \gt 0. Then, since c is the least upper bound and c - \epsilon \lt c , c - \epsilon does not bound sequence a_n , implying existence of m for which a_n \gt c - \epsilon. Since a_n is monotone,
meaning that |c - a_n| \lt \epsilon, which exactly means that c = \lim_{n\to\infty} a_n .
Observe that also c = sup\{a_n\} , where \sup is the least upper bound of the set \{a_n\} .