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The concept of supremum, or least upper bound, is as follows: Let <math>S={a[n]}</math>, the sequence with terms <math>a[0],a[1],\cdots</math> over all the nonnegative integers. <math>S</math> has a supremum, called <math>\sup S</math> , if for every <math>n , a[n]\leq\sup S</math> (i.e. no a[n] exceeds <math>\sup S</math> ), and furthermore, <math>\sup S</math> is the least value with this property; that is, if <math>a[n]\leq b</math> for all <math>n</math>, then <math>\sup S\leq b</math> for all such <math>b</math> . This is why the supremum is also called the least upper bound, for a bound is a number which a function, sequence, or set, never exceeds. Similarly, one can define the infimum <math>\inf S</math> , or greatest lower bound. | The concept of supremum, or least upper bound, is as follows: Let <math>S={a[n]}</math>, the sequence with terms <math>a[0],a[1],\cdots</math> over all the nonnegative integers. <math>S</math> has a supremum, called <math>\sup S</math> , if for every <math>n , a[n]\leq\sup S</math> (i.e. no a[n] exceeds <math>\sup S</math> ), and furthermore, <math>\sup S</math> is the least value with this property; that is, if <math>a[n]\leq b</math> for all <math>n</math>, then <math>\sup S\leq b</math> for all such <math>b</math> . This is why the supremum is also called the least upper bound, for a bound is a number which a function, sequence, or set, never exceeds. Similarly, one can define the infimum <math>\inf S</math> , or greatest lower bound. | ||
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| + | • Consider the set <math>\left\{ x:\;0<x<1\right\}</math> . There is no maximum or minimum, however <math>0</math> is the infimum and <math>1</math> is the supremum. | ||
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| + | • Consider the set <math>S={a[n]},\; a[n]=1/n</math> where <math>n</math> is a positive integer. | ||
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| + | – <math>\sup S=1</math> , since <math>1/n>1/(n+1)</math> for all such <math>n</math> , and so the largest term is the first. The maximum is also <math>1</math>. | ||
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| + | – <math>\inf S=0</math> . However, the minimum does not exist. | ||
Revision as of 11:20, 16 November 2010
1.1 Basic Mathematics
1.1.1 Mathematical notation
• ≈ : approximately equal
• ~ : CST ·
Supremum and infimum vs. maximum and minimum
The concept of supremum, or least upper bound, is as follows: Let $ S={a[n]} $, the sequence with terms $ a[0],a[1],\cdots $ over all the nonnegative integers. $ S $ has a supremum, called $ \sup S $ , if for every $ n , a[n]\leq\sup S $ (i.e. no a[n] exceeds $ \sup S $ ), and furthermore, $ \sup S $ is the least value with this property; that is, if $ a[n]\leq b $ for all $ n $, then $ \sup S\leq b $ for all such $ b $ . This is why the supremum is also called the least upper bound, for a bound is a number which a function, sequence, or set, never exceeds. Similarly, one can define the infimum $ \inf S $ , or greatest lower bound.
• Consider the set $ \left\{ x:\;0<x<1\right\} $ . There is no maximum or minimum, however $ 0 $ is the infimum and $ 1 $ is the supremum.
• Consider the set $ S={a[n]},\; a[n]=1/n $ where $ n $ is a positive integer.
– $ \sup S=1 $ , since $ 1/n>1/(n+1) $ for all such $ n $ , and so the largest term is the first. The maximum is also $ 1 $.
– $ \inf S=0 $ . However, the minimum does not exist.
