Monotone convergence theorem

In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the convergence of monotonic sequences (sequences that are increasing or decreasing) that are also bounded. Informally, the theorems state that if a sequence is increasing and bounded above by a supremum, then the sequence will converge to the supremum; in the same way, if a sequence is decreasing and is bounded below by an infimum, it will converge to the infimum. In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the convergence of monotonic sequences (sequences that are increasing or decreasing) that are also bounded. Informally, the theorems state that if a sequence is increasing and bounded above by a supremum, then the sequence will converge to the supremum; in the same way, if a sequence is decreasing and is bounded below by an infimum, it will converge to the infimum. If a sequence of real numbers is increasing and bounded above, then its supremum is the limit. Let { a n } {displaystyle {a_{n}}} be such a sequence. By assumption, { a n } {displaystyle {a_{n}}} is non-empty and bounded above. By the least-upper-bound property of real numbers, c = sup n { a n } {displaystyle c=sup _{n}{a_{n}}} exists and is finite. Now, for every ε > 0 {displaystyle varepsilon >0} , there exists N {displaystyle N} such that a N > c − ε {displaystyle a_{N}>c-varepsilon } , since otherwise c − ε {displaystyle c-varepsilon } is an upper bound of { a n } {displaystyle {a_{n}}} , which contradicts to the definition of c {displaystyle c} . Then since { a n } {displaystyle {a_{n}}} is increasing, and c {displaystyle c} is its upper bound, for every n > N {displaystyle n>N} , we have | c − a n | ≤ | c − a N | < ε {displaystyle |c-a_{n}|leq |c-a_{N}|<varepsilon } . Hence, by definition, the limit of { a n } {displaystyle {a_{n}}} is sup n { a n } . {displaystyle sup _{n}{a_{n}}.} If a sequence of real numbers is decreasing and bounded below, then its infimum is the limit.