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Difference between revisions of "Increasing sequence"

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A sequence of real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050510/i0505101.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050510/i0505102.png" /> the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050510/i0505103.png" /> is satisfied. Such a sequence is sometimes called a strictly-increasing sequence, the term  "increasing sequence"  being reserved for sequences that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050510/i0505104.png" /> merely satisfy the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050510/i0505105.png" />. Such sequences are also known as non-decreasing sequences. All non-decreasing sequences which are bounded from above have a finite limit, while all those not bounded from above have an infinite limit, equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050510/i0505106.png" />.
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A sequence of real numbers $x_n$ such that for all $n=1,2,\ldots,$ the inequality $x_n<x_{n+1}$ is satisfied. Such a sequence is sometimes called a strictly-increasing sequence, the term  "increasing sequence"  being reserved for sequences that for all $n$ merely satisfy the inequality $x_n\leq x_{n+1}$. Such sequences are also known as non-decreasing sequences. All non-decreasing sequences which are bounded from above have a finite limit, while all those not bounded from above have an infinite limit, equal to $+\infty$.
  
  

Latest revision as of 16:28, 1 May 2014

A sequence of real numbers $x_n$ such that for all $n=1,2,\ldots,$ the inequality $x_n<x_{n+1}$ is satisfied. Such a sequence is sometimes called a strictly-increasing sequence, the term "increasing sequence" being reserved for sequences that for all $n$ merely satisfy the inequality $x_n\leq x_{n+1}$. Such sequences are also known as non-decreasing sequences. All non-decreasing sequences which are bounded from above have a finite limit, while all those not bounded from above have an infinite limit, equal to $+\infty$.


Comments

See also Decreasing sequence.

How to Cite This Entry:
Increasing sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Increasing_sequence&oldid=14839
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article