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.