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- ...n \in \mathbb{Z}_{> 0}\right\}</math>. While this set has a [[least upper bound]] 1, it has no maximum. ...operty]], then <math>S</math> has a maximum if and only if the least upper bound of <math>S</math> is a member of <math>S</math>.1 KB (232 words) - 11:55, 9 February 2007
- ...ery <math>s \in S</math> and there is no <math>m < M</math> with this same property. If the least upper bound <math>M</math> of <math>S</math> is an element of <math>S</math>, it is als1,011 bytes (177 words) - 13:08, 5 March 2022
- ...ery <math>s \in S</math> and there is no <math>M > m</math> with this same property. If the greatest lower bound <math>m</math> of <math>S</math> is an element of <math>S</math>, it is als597 bytes (109 words) - 12:55, 5 March 2022
- ...'' is actually a trivial corollary of the [[Least upper bound|completeness property]] of <math>\mathbb{R}</math> but is extremely useful in real analysis We see that <math>u'</math> is an upper bound of <math>A</math>, but <math>u'<u</math> which contradicts the assumption t700 bytes (118 words) - 02:44, 15 February 2008
- ...y [[filter]] <math>\mathcal{F}</math> on <math>X</math> with the following property: ...lter is said to be ''principle'', or ''fixed'', or ''trivial'' if it has a least element, i.e., an element which is a subset of all the others. Otherwise,9 KB (1,685 words) - 19:28, 13 October 2019
- First we prove the lower bound. Note that, by the [[Pigeonhole Principle]], at least two of <math>a,b,c</math> are either both greater than or less than <math>14 KB (807 words) - 09:45, 9 April 2023
- Specifically, it is a module <math>M</math> with the following property: Then <math>\bigcup_{i\in I} N_i</math> is an upper bound on the chain6 KB (1,183 words) - 14:02, 18 August 2009
- ...t most <math>20</math>. How many points with integer coordinates lie on at least one of these paths? ...s that the bug has to move. If the point <math>(x, y)</math> satisfies the property that <math>|x - 3| + |y + 2| + |x + 3| + |y - 2| \le 20</math>, then it is7 KB (1,206 words) - 15:28, 5 September 2024
- then <math>M</math> is an upper bound of <math>S</math>. If there exists a number <math>m</math> such that such t then <math>m</math> is a lower bound of <math>S</math>. We may then say:18 KB (2,905 words) - 17:33, 5 April 2023
- ...rs). This property of the real numbers is known as the ''least upper bound property''.9 KB (1,409 words) - 01:41, 30 May 2023
- ...all cases gets <math>15+13+7+1=36</math>. However, due to the commutative property, we must multiply this by 2. For example, in the <math>2,17,19</math> case ...the upper bound for each case increases by <math>1</math>, but commutative property doubles it to be <math>4</math>.3 KB (481 words) - 06:35, 25 July 2024