limit of nondecreasing sequence

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

Proof.  Let  a1a2anM.  Therefore the set  {a1,a2,} has a finite supremum sM.  We show that

limnan=s. (1)

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

0s-ans-an(ε)<εfor allnn(ε).

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