Difference between revisions of "Decreasing sequence"
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| − | A sequence $\{x_n\}$ such that for each $n=1,2,\ldots,$ one has $x_n>x_{n+1}$. Sometimes such a sequence is called strictly decreasing, while the term "decreasing sequence" is applied to a sequence satisfying for all $n$ the condition $x_n\geq x_{n+1}$. Such a sequence is sometimes called non-increasing. Every non-increasing sequence that is bounded from below has a finite limit, while one that is not bounded from below has limit $-\infty$. | + | A sequence $\{x_n\}$ such that for each $n=1,2,\ldots,$ one has $x_n>x_{n+1}$. Sometimes such a sequence is called strictly decreasing, while the term "decreasing sequence" is applied to a sequence satisfying for all $n$ the condition $x_n\geq x_{n+1}$. Such a sequence is sometimes called non-increasing. |
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| + | Every non-increasing sequence of real numbers that is bounded from below has a finite limit, while one that is not bounded from below has limit $-\infty$. See [[Continuity axiom]]. | ||
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| + | [[Category:Order, lattices, ordered algebraic structures]] | ||
Latest revision as of 21:39, 25 October 2014
A sequence $\{x_n\}$ such that for each $n=1,2,\ldots,$ one has $x_n>x_{n+1}$. Sometimes such a sequence is called strictly decreasing, while the term "decreasing sequence" is applied to a sequence satisfying for all $n$ the condition $x_n\geq x_{n+1}$. Such a sequence is sometimes called non-increasing.
Every non-increasing sequence of real numbers that is bounded from below has a finite limit, while one that is not bounded from below has limit $-\infty$. See Continuity axiom.
How to Cite This Entry:
Decreasing sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Decreasing_sequence&oldid=32102
Decreasing sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Decreasing_sequence&oldid=32102
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article