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,

\forall n > m: c - \epsilon < a_n \le c,

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\} .