limit of nondecreasing sequence

Theorem. A monotonically nondecreasing sequence of real numbers with upper bound a number $M$ converges to a limit which does not exceed $M$ .

Proof. Let $a_{1}\leqq a_{2}\leqq\ldots\leqq a_{n}\leqq\ldots\leqq M$ . Therefore the set $\{a_{1},\,a_{2},\,\ldots\}$ has a finite supremum $s\leqq M$ . We show that

\displaystyle\lim_{n\to\infty}a_{n}\;=\;s.

(1)

Let $\varepsilon$ an arbitrary positive number. According to the definition of supremum we have $a_{n}\leqq s$ for all $n$ and on the other hand, there exists a member $a_{n(\varepsilon)}$ of the sequence that is $>s-\varepsilon$ . Then we have $s-\varepsilon<a_{n(\varepsilon)}\leqq s$ , and since the sequence is nondecreasing,

0\;\leqq\;s-a_{n}\;\leqq\;s\!-\!a_{n(\varepsilon)}\;<\;\varepsilon\quad\mbox{% for all}\;\,n\geqq n(\varepsilon).

Thus the equation (1) and the whole theorem has been proven.

For the nonincreasing sequences there is the corresponding

Theorem. A monotonically nonincreasing sequence of real numbers with lower bound a number $L$ converges to a limit which is not less than $L$ .

Note. A good application of the latter theorem is in the proof that Euler’s constant exists.

Title	limit of nondecreasing sequence
Canonical name	LimitOfNondecreasingSequence
Date of creation	2013-03-22 17:40:31
Last modified on	2013-03-22 17:40:31
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	13
Author	pahio (2872)
Entry type	Theorem
Classification	msc 40-00
Synonym	nondecreasing sequence with upper bound
Synonym	limit of increasing sequence
Related topic	MonotonicallyIncreasing
Related topic	MonotoneIncreasing
Related topic	Supremum
Related topic	Infimum
Related topic	ConvergenceOfTheSequence11nn